Some basic topics in analysis related to shift operators

Stephen Semmes Rice University

Abstract Some topics related to shift operators are discussed, in connection with absolute value functions on fields, and norms on vector spaces over such fields.

Contents

I Basic notions 4

1 Vector spaces and linear mappings 4 2 k-Valued functions 5 3 Bilateral shift operators 7 4 Nonnegative sums 8 5 q-Semimetrics 10 6 q-Absolute value functions 12 7 q-Seminorms 15 8 r-Summable functions 16 9 Bounded linear mappings 18 10 Summable functions 21 11 Hilbert spaces 22 12 Infinite series 24 13 Completeness 25 14 Orthonormal systems 27 15 Fourier series 28 16 Fourier series, continued 30 17 Some bounded linear functionals 32 18 Some bounded linear mappings 34 19 Multiplication operators 36 20 Dual linear mappings 37 21 Hilbert space adjoints 38

II Invertibility and related topics 41

22 Invertibility of multiplication operators 41 23 The usual series 42 24 Some consequences 44 25 Eigenvalues and eigenvectors 46 26 Eigenfunctions for bilateral shifts 48 27 Finite support in Z 49 28 Arbitrary functions on Z 51 29 Summable functions on Z 53 30 Approximate eigenvalues 55 31 Eigenvalues of multiplication operators 58 32 Eigenvalues of bilateral shifts 60 33 The ultrametric case 62 34 k Complete 63 35 The domain of R 66 36 Doubly-infinite sums 67 37 Bounded partial sums 69 38 Normalized partial sums 70 39 Connections with I − T 73 40 Some density conditions 74 41 Limits of partial sums 76 42 The inverse of I − a T 78

III Unilateral shift operators 79

43 Forward and backward shifts 79 44 Polynomials and power series 81 45 Extension and restriction mappings 82 46 Dual mappings 84 47 Boundedness on ℓ^{r} spaces 85 48 Adjoint mappings 87 49 Eigenfunctions for unilateral shifts 88 50 Eigenvalues of unilateral shifts 90 51 The case where q = r = ∞ 92 52 Multiplicative inverses in k[[X]] 93 53 Inverting I − a A on c(Z_{0+}, k) 95 54 ℓ^{r} Estimates, r ≤ q, r < ∞ 96 55 Estimates for k = R, C 98 56 Convergent power series 99 57 Hardy spaces 100 References 102

Index 105

Part I

Basic notions

1 Vector spaces and linear mappings

Let k be a field, and let V, W be vector spaces over k. The space of linear mappings from V into W will be denoted L(V, W), and is a vector space over k with respect to pointwise addition and scalar multiplication. If V = W, then this space may be denoted L(V). This is an associative algebra over k, with composition of linear mappings as multiplication. The identity mapping from V into itself may be denoted I or I_{V}, which is the multiplicative identity element in L(V).

As usual, a linear functional on V is a linear mapping from V into k, where k is considered as a one-dimensional vector space over itself. The space of linear functionals on V is known as the algebraic dual of V, which may be denoted V^{alg}. This is the same as L(V, k), using the notation in the preceding paragraph, and in particular this is a vector space over k. If V has finite dimension, then it is well known that V^{alg} has the same dimension. A version of this will be discussed in the next section.

Let W be another vector space over k, and let T be a linear mapping from V into W. If λ is a linear functional on W, then the composition λ ∘ T of λ and T defines a linear functional on V. Put

T^{alg}(λ) = λ ∘ T, (1.1)

which defines a linear mapping from W^{alg} into V^{alg}, dual to T. Note that

T ↦ T^{alg} (1.2)

defines a linear mapping from L(V, W) into L(W^{alg}, V^{alg}). If V = W, then we can apply this to T = I_{V}, and it is easy to see that

(I_{V})^{alg} = I_{V^{alg}}, (1.3)

the identity mapping on V^{alg}.

Let V, W, and Z be vector spaces over k, let T_{1} be a linear mapping from V into W, and let T_{2} be a linear mapping from W into Z, so that the composition T_{2} ∘ T_{1} is a linear mapping from V into W. If λ is a linear functional on Z, then

(1.4) (T_{2}∘T_{1})^{alg}(λ) = λ∘(T_{2}∘T_{1}) = (λ∘T_{2})∘T_{1} = (T_{2}^{alg}(λ))∘T_{1} = T_{1}^{alg}(T_{2}^{alg}(λ)).

This implies that (1.5) (T_{2} ∘ T_{1})^{alg} = T_{1}^{alg} ∘ T_{2}^{alg} as linear mappings from V_{3}^{alg} into V_{1}^{alg}. Suppose now that T is a one-to-one linear mapping from V onto W, and let T^{−1} be the corresponding inverse mapping from W onto V. Under these conditions, T^{alg} is a one-to-one linear mapping from W^{alg} onto V^{alg}, with

(T^{alg})^{−1} = (T^{−1})^{alg} (1.6)

as linear mappings from V^{alg} onto W^{alg}.

Let V be a vector space over k again, and put

L_{v}(λ) = L_{V,v}(λ) = λ(v) (1.7)

for each v ∈ V and λ ∈ V^{alg}. This defines a linear functional on V^{alg} for each v ∈ V, which is to say an element of (V^{alg})^{alg}. Moreover,

v ↦ L_{v} (1.8)

defines a linear mapping from V into (V^{alg})^{alg}. Let W be another vector space over k, and let T be a linear mapping from V into W, so that T^{alg} is defined as a linear mapping from W^{alg} into V^{alg} as before. If v ∈ V, then L_{V,v} ∘ T^{alg} is defined as a linear functional on W^{alg}, so that (L_{V,v} ∘ T^{alg})(μ) is defined as an element of k for each μ ∈ W^{alg}. Observe that

(1.9) (L_{V,v} ∘ T^{alg})(μ) = L_{V,v}(T^{alg}(μ)) = L_{V,v}(μ ∘ T)
= (μ ∘ T)(v) = μ(T (v)) = L_{W,T (v)}(μ)

for every v ∈ V and μ ∈ W^{alg}, where L_{W,w} is defined as an element of (W^{alg})^{alg} for every w ∈ W as in (1.7). Thus

L_{v} ∘ T^{alg} = L_{W,T (v)} (1.10)

as linear functionals on W^{alg} for every v ∈ V. Note that (T^{alg})^{alg} can be defined as a linear mapping from (V^{alg})^{alg} into (W^{alg})^{alg} as before. Using (1.10), we get that (1.11) (T^{alg})^{alg}(L_{V,v}) = L_{V,v} ∘ T^{alg} = L_{W,T (v)}

for each v ∈ V.

2 k-Valued functions

Let k be a field, and let X be a nonempty set. The space of k-valued functions on X will be denoted c(X, k), which is a vector space over k with respect to pointwise addition and scalar multiplication. The support of a k-valued function f on X is defined by

supp f = {x ∈ X : f(x) ≠ 0}. (2.1)

The space of k-valued functions on X whose support have only finitely many elements is denoted c_{00}(X, k), and is a linear subspace of c(X, k). If y ∈ X, then δ_{y} = δ_{X,y} is defined as a k-vaued function on X by

(2.2)  δ_{y}(x) = δ_{X,y}(x) = 1 when x = y
= 0 when x ≠ y.

Thus δ_{y} ∈ c_{00}(X, k) for each y ∈ X, and

{δ_{y} : y ∈ X} (2.3)

is a basis for c_{00}(X, k) as a vector space over k. Of course, if X has only finitely many elements, then c_{00}(X, k) is the same as c(X, k).

If f ∈ c_{00}(X, k), then

∑_{x∈X} f(x) (2.4)

reduces to a finite sum in k, and thus defines an element of k. The mapping from f ∈ c_{00}(X, k) to the sum (2.4) defines a linear functional on c_{00}(X, k).

Let g ∈ c(X, k) be given. If f ∈ c_{00}(X, k), then the product f g of f and g has finite support in X as well. Thus

λ_{g}(f) = ∑_{x∈X} f(x) g(x) (2.5)

defines an element of k, as in the preceding paragraph. Hence (2.5) defines a linear functional on c_{00}(X, k). Note that

λ_{g}(δ_{y}) = g(y) (2.6)

for every y ∈ X. If λ is any linear functional on c_{00}(X, k), then we can put

g_{λ}(y) = λ(δ_{y}) (2.7)

for each y ∈ X. This defines a k-valued function on X, and one can check that

λ = λ_{g_{λ}} (2.8)

as linear functionals on c_{00}(X, k). It follows that g ↦ λ_{g} defines a one-to-one linear mapping from c(X, k) onto c_{00}(X, k)^{alg}, with inverse given by λ ↦ g_{λ}.

Similarly, if h ∈ c_{00}(X, k), then

μ_{h}(g) = ∑_{x∈X} g(x) h(x) (2.9)

defines a linear functional on c(X, k), where the sum on the right side of (2.9) also reduces to a finite sum in k. The restriction of μ_{h} to c_{00}(X, k) is the same as λ_{h} as defined in the preceding paragraph. In particular,

μ_{h}(δ_{y}) = h(y) (2.10) for every y ∈ X, so that h ↦ μ_{h} defines a one-to-one linear mapping from c_{00}(X, k) into c(X, k)^{alg}. If f ∈ c_{00}(X, k), then we can define L_{f} as a linear functional on c_{00}(X, k)^{alg} as in (1.7). This linear functional is given by

L_{f}(λ_{g}) = λ_{g}(f) = μ_{f}(g) (2.11)

for each g ∈ c(X, k), where λ_{g} is as in (2.5).

3 Bilateral shift operators

Let k be a field, and let c(Z, k) be the space of k-valued mappings from the set Z of integers into k, as in the previous section. Consider the linear mapping T from c(Z, k) into itself defined by

(T (f))(j) = f(j − 1) (3.1)

for every f ∈ c(Z, k) and j ∈ Z. This is the standard forward shift operator on c(Z, k), which is a one-to-one linear mapping from c(Z, k) onto itself. Note that T maps c_{00}(Z, k) onto itself. The inverse of T is given by

(T^{−1}(f))(j) = f(j + 1) (3.2)

for each f ∈ c(Z, k) and j ∈ Z, which is the standard backward shift operator on c(Z, k).

Let T^{l} be the lth power of T as a linear mapping from c(Z, k) into itself with respect to composition for each positive integer l. If l = 0, then T^{l} is interpreted as being the identity mappping on c(Z, k). In this situation, we can also define T^{l} for negative integers l, by taking powers of T^{−1} on c(Z, k). Thus T^{l} is defined as a linear mapping on c(Z, k) for every integer l, and it is easy to see that (T^{l}(f))(j) = f(j − l) (3.3)

for every f ∈ c(Z, k) and j, l ∈ Z.

Let δ_{n} = δ_{Z,n} be the k-valued function defined on Z as in (2.2) for each n ∈ Z. Observe that

(T^{l}(δ_{n}))(j) = δ_{n}(j − l) = δ_{n+l}(j) (3.4)

for every j, l, n ∈ Z. Thus T^{l}(δ_{n}) = δ_{n+l} (3.5)

for every l, n ∈ Z. If A ⊆ Z and l ∈ Z, then let A + l be the subset of Z defined by A + l = {a + l : a ∈ A}. (3.6)

Using (3.3), we get that

(3.7) supp T^{l}(f) = (supp f) + l for every f ∈ c(Z, k) and l ∈ Z.

If g ∈ c(Z, k), then

λ_{g}(f) = ∑_{j=−∞}^{∞} f(j) g(j) (3.8)

defines a linear functional on c_{00}(Z, k), as in (2.5). Observe that

λ_{g}(T (f)) = ∑_{j=−∞}^{∞} f(j − 1) g(j) = ∑_{j=−∞}^{∞} f(j) g(j + 1) = λ_{T^{−1}(g)}(f) (3.9)

for every f ∈ c_{00}(Z, k) and g ∈ c(Z, k). This says that the algebraic dual of T as a linear mapping from c_{00}(Z, k) into itself corresponds to T^{−1} as a linear mapping from c(Z, k) into itself. This uses the identification of c_{00}(Z, k)^{alg} with c(Z, k) given by (3.8), as in the previous section. Similarly, if h ∈ c_{00}(Z, k), then

μ_{h}(g) = ∑_{j=−∞}^{∞} g(j) h(j) (3.10)

defines a linear functional on c(Z, k), as in (2.9). As before, we have that

μ_{h}(T (g)) = μ_{T^{−1}(h)}(g) (3.11)

for every g ∈ c(Z, k) and h ∈ c_{00}(Z, k). In this situation, we can identify c_{00}(Z, k) with a linear subspace of c(Z, k)^{alg}, using (3.10). With respect to this identification, the restriction of the algebraic dual of T to this subspace of c(Z, k)^{alg} corresponds to the restriction of T^{−1} to c_{00}(Z, k), by (3.11).

4 Nonnegative sums

Let f(x) be a nonnegative real-valued function on a nonempty set X. If X has only finitely many elements, or if f has finite support in X, then the sum

∑_{x∈X} f(x) (4.1)

can be defined as a nonnegative real number in the usual way. Otherwise, (4.1) can be defined as a nonnegative extended real number as the supremum of

∑_{x∈E} f(x) (4.2)

over all nonempty finite subsets E of X. If X is the set Z_{+} of positive integers, then (4.1) is the same as

∑_{j=1}^{∞} f(j) = sup_{n≥1} ∑_{j=1}^{n} f(j), (4.3)

where more precisely the supremum is taken over all positive integers n. Similarly, if X = Z, then (4.1) is the same as

∑_{j=−∞}^{∞} f(j) = sup_{l,n≥1} ∑_{j=−l}^{n} f(j) = sup_{n≥1} ∑_{j=−n}^{n} f(j). (4.4)

Let X be any nonempty set again, and let f, g be nonnegative real-valued functions on X. If X has only finitely many elements, or if f and g have finite support in X, then

∑_{x∈X}(f(x) + g(x)) = ( ∑_{x∈X} f(x)) + ( ∑_{x∈X} g(x)). (4.5)

One can check that (4.5) holds without these additional conditions, by approximating these sums by finite subsums. This also uses standard conventions for sums of nonnegative extended real numbers. Similarly,

∑_{x∈X} t f(x) = t ∑_{x∈X} f(x) (4.6)

for every nonnegative real number t when X has finitely many elements, or f has finite support in X. This also holds without these additional conditions when t > 0, where the right side is interpreted as being +∞ when (4.1) is infinite. If t = 0, then it is customary to interpret the right side as being 0 even when (4.1) is infinite.

If f is a nonnegative real-valued function on X and r is a positive real number, then we put

∥f∥_{r} = ( ∑_{x∈X} f(x)^{r})^{1/r}, (4.7)

which is interpreted as being +∞ when the sum is infinite. As usual, we can extend this to r = ∞ by putting

∥f∥_{∞} = sup_{x∈X} f(x), (4.8)

which is defined as a nonnegative extended real number. Observe that

∥f∥_{∞} ≤ ∥f∥_{r} (4.9)

for every r > 0. If 0 < r_{1} ≤ r_{2} < ∞, then we have that

∥f∥_{r_{2}}^{r_{2}} = ∑_{x∈X} f(x)^{r_{2}} ≤ ∥f∥_{∞}^{r_{2}−r_{1}} ∑_{x∈X} f(x)^{r_{1}} = ∥f∥_{∞}^{r_{2}−r_{1}} ∥f∥_{r_{1}}^{r_{1}} ≤ ∥f∥_{r_{1}}^{r_{2}}, (4.10)

using (4.9) in the last step. This implies that

∥f∥_{r_{2}} ≤ ∥f∥_{r_{1}} (4.11)

when r_{1} ≤ r_{2}, which also works when r_{2} = ∞, by (4.9).

If f, g are nonnegative real-valued functions on X and 1 < r < ∞, then it is well known that ∥f + g∥_{r} ≤ ∥f∥_{r} + ∥g∥_{r}, (4.12)

by Minkowski’s inequality for sums.. Of course, equality holds trivially when r = 1, and it is easy to check directly that this inequality also holds when r = ∞. If 0 < r ≤ 1, then (a + b)^{r} ≤ a^{r} + b^{r} (4.13)

for all nonnegative real numbers a, b. This follows from (4.11), with r_{1} = r, r_{2} = 1, and where X has exactly two elements. It follows that

(4.14)  ∥f + g∥_{r}^{r} = ∑_{x∈X}(f(x) + g(x))^{r}
≤ ∑_{x∈X}(f(x)^{r} + g(x)^{r})
= ∑_{x∈X} f(x)^{r} + ∑_{x∈X} g(x)^{r} = ∥f∥_{r}^{r} + ∥g∥_{r}^{r}

for all nonnegative real-valued functions f, g on X when 0 < r ≤ 1, using (4.13) in the second step, and (4.5) in the third step.

5 q-Semimetrics

Let X be a set, and let q be a positive real number. A nonnegative real-valued function d(x, y) defined on X ×X is said to be a q-semimetric on X if it satisfies the following three conditions. First,

(5.1) d(x, x) = 0 for every x ∈ X.

Second, (5.2) d(x, y) = d(y, x) for every x, y ∈ X.

Third, (5.3) d(x, z)^{q} ≤ d(x, y)^{q} + d(y, z)^{q} for every x, y, z ∈ X.

If we also have that d(x, y) > 0 (5.4)

for every x, y ∈ X with x ≠ y, then d(·, ·) is said to be a q-metric on X. A q-semimetric or q-metric with q = 1 is also known as a semimetric or metric, as appropriate. Note that (5.3) may be reformulated as saying that

d(x, z) ≤ (d(x, y)^{q} + d(y, z)^{q})^{1/q} (5.5)

for every x, y, z ∈ X. The right side of (5.5) decreases monotonically in q, by (4.11). This means that the property of being a q-semimetric or a q-metric becomes more restrictive as q increases.

A nonnegative real-valued function d(x, y) defined on X × X is said to be a semi-ultrametric on X if it satisfies (5.1) and (5.2), and also

d(x, z) ≤ max(d(x, y), d(y, z)) (5.6)

for every x, y, z ∈ X. If (5.4) holds as well, then d(·, ·) is said to be an ultrametric on X. Observe that the right side of (5.6) is less than or equal to the right side of (5.5), so that (5.6) implies (5.5). This means that a semi-ultrametric is a q-semimetric for each q > 0, and similarly an ultrametric is a q-metric for every q > 0. Semi-ultrametrics and ultrametrics will be considered as q-semimetrics and q-metrics with q = ∞, respectively.

The discrete metric is defined on any set X by putting d(x, y) equal to 1 when x ≠ y, and equal to 0 when x = y. It is easy to see that this defines an ultrametric on X. Let d(x, y) be any q-semimetric on a set X, 0 < q ≤ ∞, and let a be a positive real number. Under these conditions, one can check that

d(x, , y)^{a} (5.7)

defines a (q/a)-semimetric on X. Similarly, if d(x, y) is a q-metric on X, then (5.7) is a (q/a)-metric on X.

Let d(·, ·) be a q-semimetric on a set X for some q > 0 again. The open ball in X centered at x ∈ X with radius r > 0 with respect to d(·, ·) is defined as usual by B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (5.8)

Similarly, the closed ball in X centered at x ∈ X with radius r ≥ 0 with respect to d(·, ·) is defined by

\overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ M : d(x, y) ≤ r\}. (5.9)

Let a be a positive real number, so that (5.7) is a (q/a)-semimetric on X, as before. If x ∈ X, then B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (5.10)

for every r > 0, and \overline{B}_{d^{a}}(x, r) = \overline{B}_{d}(x, r) (5.11)

for every r ≥ 0.

A subset U of X is said to be an open set with respect to d(·, ·) if for every x ∈ U there is an r > 0 such that

B(x, r) ⊆ U, (5.12)

which defines a topology on X. If a is a positive real number, then (5.7) defines a (q/a)-metric on X, and the topology determined on X by (5.7) is the same as the topology determined by d(·, ·), because of (5.10). One can verify directly that the topology determined on X by d(·, ·) satisfies many of the same properties as for ordinary semimetrics, but one can also use the preceding remark to reduce to that case. In particular, open balls in X with respect to d(·, ·) are open sets, and closed balls are closed sets. If d(·, ·) is a q-metric on X, then it follows that X is Hausdorff with respect to this topology.

Suppose for the moment that d(·, ·) is a semi-ultrametric on X. If x, y ∈ X satisfy d(x, y) < r for some r > 0, then it is easy to see that

B(y, r) ⊆ B(x, r). (5.13)

It follows that B(x, r) = B(y, r) (5.14)

under these conditions, since we can interchange the roles of x and y. Similarly, if d(x, y) ≤ r for some r ≥ 0, then

\overline{B}(y, r) ⊆ \overline{B}(x, r), (5.15)

and hence \overline{B}(x, r) = \overline{B}(y, r), (5.16)

by interchanging the roles of x and y again. Using (5.15), we get that \overline{B}(x, r) is an open set in X when r > 0, and one can check that open balls in X are closed sets in this situation too.

Let d_{X} be a q_{X}-metric on X for some q_{X} > 0. One can define the notions of Cauchy sequences in X and completeness of X with respect to d_{X} in the usual way. If a is a positive real number, then d_{X}(·, ·)^{a} determines the same collection of Cauchy sequences in X, and the same notion of completeness. Similarly, if Y is another set, and d_{Y} is a q_{Y} -metric on Y for some q_{Y} > 0, then one can define the notion of uniform continuity of mappings from X into Y with respect to d_{X} and d_{Y} in the usual way. If a, b are positive real numbers, then d_{X}(·, ·)^{a} and d_{Y} (·, ·)^{b} determine the same collection of uniformly continuous mappings from X into Y.

6 q-Absolute value functions

Let k be a field, and let q be a positive real number. A nonnegative real-valued function |x| defined on k is said to be a q-absolute value function on k if it satisfies the following three conditions. First,

(6.1) |x| = 0 if and only if x = 0.

More precisely, this means that |x| = 0 as a real number if and only if x = 0 as an element of k. Second,

(6.2) |x y| = |x| |y| for every x, y ∈ k.

Third, (6.3) |x + y|^{q} ≤ |x|^{q} + |y|^{q} for every x, y ∈ k.

If these conditions hold with q = 1, then | · | is said to be an absolute value function on k. It is well known that the standard absolute value functions on the fields R and C of real and complex numbers, respectively, are absolute value functions in this sense.

Using (6.1) and (6.2), one can check that |1| = 1, where the first 1 is the multiplicative identity element in k, and the second 1 is the multiplicative identity element in R. If x ∈ k satisfies x^{n} = 1 for some positive integer n, then it follows that |x| = 1, using (6.2) again. In particular, this holds when x = −1, the additive inverse of 1 in k. This implies in turn that | − y| = |y| for every y ∈ k, using (6.2). If | · | is a q-absolute value function on k, then we get that

d(x, y) = |x − y| (6.4)

defines a q-metric on k.

As in the previous section, (6.3) is equivalent to asking that

|x + y| ≤ (|x|^{q} + |y|^{q})^{1/q} (6.5)

for every x, y ∈ k. The right side of (6.5) decreases monotonically in q, as in (4.11), so that the property of being a q-absolute value function becomes more restrictive as q increases. A nonnegative real-valued function | · | on k is said to be an ultrametric absolute value function if it satisfies (6.1), (6.2), and

|x + y| ≤ max(|x|, |y|) (6.6)

for every x, y ∈ k. This implies that | · | is a q-absolute value function on k for every q > 0, because the right side of (6.6) is less than or equal to the right side of (6.5). If | · | is an ultrametric absolute value function on k, then (6.4) is an ultrametric on k. As before, an ultrametric absolute value function on k may be considered as a q-absolute value function with q = ∞. The trivial absolute value function is defined on any field k by putting |x| = 1 when x ≠ 0 and |0| = 0. It is easy to see that this defines an ultrametric absolute value function on k, for which the corresponding ultrametric is the discrete metric.

If | · | is not the trivial absolute value function on k, then there is an x ∈ k such that x ≠ 0 and |x| ≠ 1. This implies that there are y, z ∈ k such that 0 < |y| < 1 and |z| > 1, using x and 1/x. It follows that |y^{j}| = |y|^{j} → 0 and |z^{j}| = |z|^{j} → ∞ as j → ∞ under these conditions.

If | · | is a q-absolute value function on a field k for some q > 0 and a is a positive real number, then one can verify that

|x|^{a} (6.7)

defines a (q/a)-absolute value function on k. Of course, the (q/a)-metric on k associated to (6.7) as in (6.4) is given by

|x − y|^{a}. (6.8)

This corresponds to the q-metric (6.4) associated to | · | on k as in (5.7), so that many of the remarks in the previous section are applicable in this situation. Let |·|_{1}, |·|_{2} be q_{1}, q_{2}-absolute value functions on k, respectively, for some q_{1}, q_{2} > 0. If there is a positive real number a such that

|x|_{2} = |x|_{1}^{a} (6.9)

for every x ∈ k, then | · |_{1} and | · |_{2} are said to be equivalent on k.

If p is a prime number, then the p-adic absolute value |x|_{p} is defined on the field Q of rational numbers as follows. Of course, we put |0|_{p} = 0. Otherwise, if x is a nonzero rational number, then x can be expressed as p^{j} (a/b), where a, b, j ∈ Z, a, b ≠ 0, and a, b are not integer multiples of p. In this case, we put

|x|_{p} = p^{−j}, (6.10)

and one can check that this defines an ultrametric absolute value function on Q. A famous theorem of Ostrowski implies that every q-absolute value function on Q is either trivial, or equivalent to the standard absolute value function on R, or equivalent to the p-adic absolute value function for some prime p.

Let | · | be a q-absolute value function on a field k for some q > 0. If k is not already complete with respect to the associated q-metric (6.4), then one can pass to a completion. This leads to a field k_{1} with a q-absolute value function | · |_{1}, such that k_{1} is complete with respect to the associated q-metric, and an isomorphism from k onto a subfield of k_{1}. This isomorphism should be isometric in the sense that | · | on k corresponds to | · |_{1} on the image, which implies that the associated q-metrics correspond in the same way. The image of k should also be dense in k_{1}, with respect to the q-metric associated to | · |_{1}. This completion is unique up to isometric isomorphic equivalence. If p is a prime number, then the completion Q_{p} of Q with respect to the p-adic absolute value function is known as the field of p-adic numbers.

Let k be a field with a q-absolute value function | · | for some q > 0 again. If x ∈ k and n ∈ Z_{+}, then we let n·x be the sum of n x’s in k. If there are positive integers n such that |n · 1| is arbitrarily large, then | · | is said to be archimedian on k. Otherwise, if there is a finite upper bound for |n · 1| with n ∈ Z_{+}, then | · | is said to be non-archimedian on k. It is easy to see that (n · 1)^{j} = n^{j} · 1 in k for every j, n ∈ Z_{+}, so that

|n^{j} · 1| = |(n · 1)^{j}| = |n · 1|^{j}. (6.11)

If |n · 1| > 1, then it follows that

|n^{j} · 1| = |n · 1|^{j} → ∞ as j → ∞, (6.12)

which means that |·| is archimedian on k. Equivalently, if |·| is non-archimedian on k, then it follows that |n · 1| ≤ 1 (6.13)

for every n ∈ Z_{+}. Ultrametric absolute value functions satisfy (6.13), and hence are non-archimedian. Conversely, it is well known that non-archimedian q-absolute value functions are ultrametric absolute value functions.

Let | · | be a q-absolute value function on a field k for some q > 0 again. Observe that {|x| : x ∈ k, x ≠ 0} (6.14)

is a subgroup of the multiplicative group R_{+} of positive real numbers. If 1 is not a limit point of (6.14) with respect to the standard topology on R, then | · | is said to be discrete on k. In this case, one can check that (6.14) has no limit points in R_{+} with respect to the standard topology on R. However, 0 is a limit point of (6.14) in R when | · | is nontrivial on k.

In order to be more precise, put

ρ_{1} = sup{|x| : x ∈ k, |x| < 1}. (6.15)

Thus 0 ≤ ρ_{1} ≤ 1 automatically. It is easy to see that ρ_{1} < 1 if and only if | · | is discrete on k, and that ρ_{1} > 0 if and only if | · | is nontrivial on k. If | · | is discrete and nontrivial on k, then the supremum in (6.15) is attained, and (6.14) consists of integer powers of ρ_{1}. If | · | is not discrete on k, then one can verify that (6.14) is dense in R_{+} with respect to the topology induced by the standard topology on R.

One can also show that | · | is non-archimedian when | · | is discrete on k. This is trivial when k has positive characteristic, and so it suffices to consider the case where k has characteristic 0. In this case, there is a natural embedding of Q into k, so that | · | induces a discrete q-absolute value function on Q. Using discreteness, it is easy to see that this induced absolute value function on Q cannot be equivalent to the standard absolute value function. Ostrowski’s theorem implies that the induced absolute value function on Q is non-archimedian, and hence that | · | is non-archimedian on k.

Let k be a field with a q-absolute value function | · | for some q > 0, and suppose that | · | is archimedian on k, and that k is complete with respect to the corresponding q-metric (6.4). Under these conditions, another theorem of Ostrowski implies that k is isomorphic to R or C, where | · | corresponds to a q-absolute value function on R or C that is equivalent to the standard one.

7 q-Seminorms

Let k be a field with a q_{k}-absolute value function for some q_{k} > 0, and let V be a vector space over k. A nonnegative real-valued function N on V is said to be a q-seminorm on V for some positive real number q and with respect to | · | on k if it satisfies the following two conditions. First,

(7.1) N(t v) = |t| N(v) for every t ∈ k and v ∈ V.

Second, (7.2) N(v + w)^{q} ≤ N(v)^{q} + N(w)^{q} for every v, w ∈ V.

Note that the first condition implies that N(0) = 0, by taking t = 0. If we also have that N(v) > 0 (7.3) for every v ∈ V with v ≠ 0, then N is said to be a q-norm on V. A q-seminorm or q-norm with q = 1 is also known as a seminorm or norm, respectively. As before, (7.2) is equivalent to asking that

N(v + w) ≤ (N(v)^{q} + N(w)^{q})^{1/q} (7.4)

for every v, w ∈ V. The right side of (7.4) decreases monotonically in q, by (4.11), so that this condition becomes more restrictive as q increases.

Similarly, a nonnegative real-valued function N on V is said to be a semiultranorm on V with respect to | · | on k if it satisfies (7.1) and

N(v + w) ≤ max(N(v), N(w)) (7.5)

for every v, w ∈ V. If N also satisfies (7.3), then N is said to be an ultranorm on V with respect to | · | on k. The right side of (7.5) is less than or equal to the right side of (7.4), so that (7.5) implies (7.4). Thus semi-ultranorms and ultranorms are automatically q-seminorms and q-norms for every q > 0, respectively. As usual, semi-ultranorms and ultranorms will be considered as q-seminorms and q-norms with q = ∞, respectively. If N is a q-seminorm or q-norm on V for some q > 0, then it is easy to see that

d(v, w) = d_{N}(v, w) = N(v − w) (7.6)

defines a q-semimetric or q-metric on V, as appropriate. Of course, | · | may be considered as a q_{k}-norm on k, where k is considered as a one-dimensional vector space over itself.

If N is a q-seminorm on V with respect to | · | on k for some q > 0, and if N(v_{0}) > 0 for some v_{0} ∈ V, then one can check that | · | has to be a q-absolute value function on k, using (7.1). More precisely, this is trivial when q ≤ q_{k}, and otherwise one can get the stronger version of the triangle inequality for | · | from the one for N when q > q_{k}. If | · | is the trivial absolute value function on k, then the trivial ultranorm is defined on V by putting N(0) = 0 and N(v) = 1 for every v ∈ V with v ≠ 0. It is easy to see that this defines an ultranorm on V, for which the corresponding ultrametric as in (7.6) is the discrete metric.

Let | · | be any q_{k}-absolute value function on k for some q_{k} > 0 again. If a is a positive real number, then | · |^{a} defines a (q_{k}/a)-absolute value function on k, as in the previous section. Similarly, if N is a q-seminorm or q-norm on V with respect to | · | on k, then N(v)^{a} (7.7)

defines a (q/a)-seminorm or (q/a)-norm on V with respect to | · |^{a} on k, as appropriate. Of course, the (q/a)-semimetric or (q/a)-metric on V associated to (7.7) as before is the same as the ath power of the q-semimetric or q-metric on V associated to N, respectively.

8 r-Summable functions

Let k be a field with a q-absolute value function | · | for some q > 0, let X be a nonempty set, and let r be a positive real number. A k-valued function f on X is said to be r-summable with respect to | · | on k if |f|^{r} is summable as a nonnegative real-valued function on X, in the sense that

∑_{x∈X} |f(x)|^{r} < ∞. (8.1)

We may also simply say that f is summable on X when this holds with r = 1, especially for k = R or C with the standard absolute value function. Let ℓ^{r}(X, k) be the space of r-summable k-valued functions on X. If f is any kvalued function on X, then

∥f∥_{r} = ∥f∥_{ℓ^{r}(X,k)} = ( ∑_{x∈X} |f(x)|^{r})^{1/r} (8.2)

is defined as a nonnegative extended real number, which is finite exactly when f is r-summable on X. Similarly, let ℓ^{∞}(X, k) be the space of k-valued functions f on X that are bounded, which means that |f| is bounded as a nonnegative real-valued function on X. As before,

∥f∥_{∞} = ∥f∥_{ℓ^{∞}(X,k)} = sup_{x∈X} |f(x)| (8.3)

is defined as a nonnegative extended real-number for any k-valued function f on X, and is finite exactly when f is bounded on X. If 0 < r_{1} ≤ r_{2} ≤ ∞, then we have that ∥f∥_{r_{2}} ≤ ∥f∥_{r_{1}} (8.4)

for every k-valued function f on X, as in (4.11). It follows that

ℓ^{r_{1}}(X, k) ⊆ ℓ^{r_{2}}(X, k) (8.5)

under these conditions, by the definition of ℓ^{r}(X, k).

If f is a k-valued function on X, t ∈ k, and t ≠ 0, then it is easy to see that

∥t f∥_{r} = |t| ∥f∥_{r} (8.6)

for every r > 0. This also holds trivially when t = 0 and f ∈ ℓ^{r}(X, k), so that the right side is defined. If g is another k-valued function on X, then we have that ∥f + g∥_{r}^{r} ≤ ∥f∥_{r}^{r} + ∥g∥_{r}^{r} (8.7)

when r ≤ q and r < ∞,

∥f + g∥_{r}^{q} ≤ ∥f∥_{r}^{q} + ∥g∥_{r}^{q} (8.8)

when q ≤ r and q < ∞, and

∥f + g∥_{∞} ≤ max(∥f∥_{∞}, ∥g∥_{∞}) (8.9)

when q = ∞. More precisely, (8.7) and (8.8) are the same when r = q < ∞, in which case they can be verified directly from the definitions. Similarly, (8.9) corresponds to r = q = ∞, and it is easy to check directly. If r ≤ q, then | · | is also an r-absolute value function on k, as in Section 6. This permits one to obtain (8.7) as in the r = q case. One can get (8.8) using Minkowski’s inequality for sums, applied to the exponent q/r ≥ 1. It follows that ℓ^{r}(X, k) is a vector space over k with respect to pointwise addition and scalar multiplication for every r > 0, and more precisely ℓ^{r}(X, k) is a linear subspace of the space c(X, k) of all k-valued functions on X. We also get that ∥f∥_{r} is an r-norm on ℓ^{∞}(X, k) when r ≤ q, and that ∥f∥_{r} is a q-norm on ℓ^{r}(X, k) when r ≥ q. Of course, the space c_{00}(X, k) of k-valued functions with finite support on X is contained in ℓ^{r}(X, k) for every r > 0. If r < ∞, then one can check that c_{00}(X, k) is dense in ℓ^{r}(X, k) with respect to the r or q-metric associated to ∥f∥_{r}, as appropriate. If | · | is the trivial absolute value function on k, then ℓ^{r}(X, k) is equal to c_{00}(X, k) when r < ∞, and ℓ^{∞}(X, k) is the same as c(X, k).

A k-valued function f on X is said to vanish at infinity if for each ϵ > 0 we have that |f(x)| < ϵ (8.10)

for all but finitely many x ∈ X. The space of k-valued functions on X that vanish at infinity is denoted c_{0}(X, k). This is a linear subspace of ℓ^{∞}(X, k), which is a closed set with respect to the q-metric associated to ∥f∥_{∞}. It is easy to see that ℓ^{r}(X, k) ⊆ c_{0}(X, k) (8.11)

when 0 < r < ∞, and in particular c_{00}(X, k) is contained in c_{0}(X, k). We also have that c_{00}(X, k) is dense in c_{0}(X, k) with respect to the q-metric associated to ∥f∥_{∞}, and the two spaces are the same when | · | is the trivial absolute value function on k.

Suppose now that k is complete with respect to the q-metric associated to | · |. Using standard arguments, one can check that ℓ^{r}(X, k) is complete for every r > 0, with respect to the r or q-metric associated to ∥f∥_{r}, as appropriate. More precisely, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in ℓ^{r}(X, k) for some r > 0, then {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in k for every x ∈ X. If k is complete, then it follows that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) of k for each x ∈ X. One can verify that f ∈ ℓ^{r}(X, k) under these conditions, and that {f_{j}}_{j=1}^{∞} converges to f with respect to ∥ · ∥_{r}.

9 Bounded linear mappings

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k. Also let N_{V}, N_{W} be q_{V}, q_{W} -norms on V, W, respectively, for some q_{V} , q_{W} > 0, and with respect to | · | on k. A linear mapping T from V into W is said to be bounded with respect to N_{V} and N_{W} if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (9.1) for every v ∈ V. This implies that

N_{W} (T (v) − T (v′)) = N_{W} (T (v − v′)) ≤ C N_{V} (v − v′) (9.2)

for every v, v′ ∈ V. In particular, it follows that T is uniformly continuous with respect to the q_{V}, q_{W} -metrics on V, W associated to N_{V}, N_{W}, respectively.

Suppose for the moment that |·| is not the trivial absolute value function on k. If a linear mapping T from V into W is continuous at 0, then one can check that T is bounded in the sense of (9.1). More precisely, it suffices to ask that N_{W} (T (v)) be bounded on a ball in V with respect to N_{V} centered at 0 with positive radius. This uses the fact that there are t ∈ k such that |t| is within a bounded factor of any positive real number, and the homogeneity property (7.1) of q-norms.

The space of bounded linear mappings from V into W is denoted BL(V, W), or simply BL(V) when V = W and N_{V} = N_{W}. If T is a bounded linear mapping from V into W, then we put

(9.3) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (9.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C for which (9.1) holds. It is easy to see that the infimum is always attained, so that (9.1) holds with C = ∥T ∥_{op}. If t ∈ k, then t T is a bounded linear mapping from V into W too, and ∥t T ∥_{op} = |t| ∥T ∥_{op}. (9.4)

This uses the analogous homogeneity properties of N_{V} and N_{W}.

Suppose that T_{1}, T_{2} are bounded linear mappings from V into W, and let C_{1}, C_{2} be nonnegative real numbers such that

N_{W} (T_{j}(v)) ≤ C_{j} N_{V} (v) (9.5)

for every v ∈ V and j = 1, 2. This implies that

N_{W} (T_{1}(v) + T_{2}(v))^{q_{W}} ≤ (C_{1}^{q_{W}} + C_{2}^{q_{W}}) N_{V} (v)^{q_{W}} (9.6)

for every v ∈ V when q_{W} < ∞, and that

N_{W} (T_{1}(v) + T_{2}(v)) ≤ max(C_{1}, C_{2}) N_{V} (v) (9.7)

for every v ∈ V when q_{W} = ∞. In both cases, it follows that T_{1} +T_{2} is bounded as a linear mapping from V into W. Thus BL(V, W) is a vector space over k with respect to pointwise addition and scalar multiplication, which is a linear subspace of the space L(V, W) of all linear mappings from V into W. It also follows easily from this that (9.3) defines a q_{W} -norm on BL(V, W).

Let T be a linear mapping from V into W, and consider the following condition on T:

(9.8) N_{W} (T (v)) is bounded for v ∈ V with N_{V} (v) ≤ 1.

This means that sup{N_{W} (T (v)) : v ∈ V, N_{V} (v) ≤ 1} (9.9)

is defined as a nonnegative real number. If T is a bounded linear mapping from V into W, then (9.8) holds, and (9.9) is less than or equal to ∥T ∥_{op}. If | · | is nontrivial on k, then (9.8) implies that T is bounded as a linear mapping from V into W, as before. In this case, one can verify that (9.1) holds with C equal to a constant multiple of (9.9), where the extra constant factor depends on | · |. It follows that ∥T ∥_{op} is less than or equal to a constant multiple of (9.9) in this situation. If k = R or C with the standard absolute value function, then this constant multiple is equal to 1, so that ∥T ∥_{op} is equal to (9.9). This also works when | · | is not discrete on k, for any field k.

Let Z be another vector space over k, and let N_{Z} be a q_{Z}-norm on Z with respect to | · | on k, for some q_{Z} > 0. Suppose that T_{1} is a bounded linear mapping from V into W, and that T_{2} is a bounded linear mapping from W into Z. Thus there are nonnegative real numbers C_{1}, C_{2} such that

N_{W} (T_{1}(v)) ≤ C_{1} N_{V} (v) (9.10)

for every v ∈ V, and N_{Z}(T_{2}(w)) ≤ C_{2} N_{W} (w) (9.11)

for every w ∈ W. Combining these two statements, we get that

N_{Z}(T_{2}(T_{1}(v))) ≤ C_{2} N_{W} (T_{1}(v)) ≤ C_{1} C_{2} N_{V} (v) (9.12)

for every v ∈ V. Hence the composition T_{2} ∘ T_{1} of T_{1} and T_{2} is bounded as a linear mapping from V into W, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (9.13)

In particular, BL(V) is closed under composition of mappings. Note that the identity mapping I_{V} on V is bounded as a linear mapping from V into itself, with ∥I_{V} ∥_{op} = 1 when V ≠ {0}.

If T is a one-to-one linear mapping from V onto W, then the inverse T^{−1} of T is a linear mapping from W onto V. A one-to-one bounded linear mapping T from V onto W is considered to be invertible as a bounded linear mapping if T^{−1} is bounded as a linear mapping from W onto V. As before, the boundedness of the inverse means that

N_{V} (T^{−1}(w)) ≤ C′ N_{W} (w) (9.14)

for some C′ ≥ 0 and every w ∈ W. In this situation, this is the same as saying that N_{V} (v) ≤ C′ N_{W} (T (v)) (9.15)

for every v ∈ V. Note that (9.15) implies that the kernel of T is trivial, so that T is injective.

A linear mapping T from V into W is said to be an isometry if

N_{W} (T (v)) = N_{V} (v) (9.16)

for every v ∈ V. This is the same as saying that (9.1) and (9.15) hold with C, C′ = 1. If T is an isometric linear mapping from V onto W, then the inverse mapping T^{−1} is an isometry from W onto V. Of course, the identity mapping I_{V} is an isometric linear mapping from V onto itself. The shift operator T discussed in Section 3 defines an isometry from ℓ^{r}(Z, k) onto itself for every r > 0, and T maps c_{0}(Z, k) onto itself as well.

Remember that k may be considered as a one-dimensional vector space over itself, and that | · | may be considered as a q_{k}-norm on k as a vector space over itself. A bounded linear functional on V is a linear functional λ on V that is bounded as a linear mapping from V into k, with respect to | · | as a norm on k. The corresponding q_{k}-operator norm of λ is known as the dual q_{k}-norm of λ.

If W is complete with respect to the q_{W} -metric associated to N_{W}, then BL(V, W) is complete with respect to the q_{W} -metric associated to (9.3), by standard arguments. More precisely, if {T_{j}}_{j=1}^{∞} is a Cauchy sequence in BL(V, W) with respect to this q_{W} -metric, then {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W for every v ∈ V, with respect to the q_{W} -metric associated to N_{W}. Hence this sequence converges in W for every v ∈ V, because W is complete, by hypothesis. It is easy to see that the limit defines a linear mapping T from V into W. One can verify that T is bounded, and that {T_{j}}_{j=1}^{∞} converges to T with respect to the operator q_{W} -norm, as desired.

10 Summable functions

In this section, we take k = R or C, with the standard absolute value function. Let X be a nonempty set, and let f be a real or complex-valued summable function on X, so that ∑_{x∈X} |f(x)| < ∞. It is well known that one can define the sum

∑_{x∈X} f(x) (10.1)

as a real or complex number, as appropriate, in a natural way under these conditions. More precisely, if f is a nonnegative real-valued function on X, then this sum can be defined as in Section 4. Otherwise, if f is a real or complexvalued summable function on X, then f can be expressed as a linear combination of nonnegative real-valued summable functions on X, which permits one to reduce to the previous case. This is a bit simpler in the real case, where f can be expressed as a difference of nonnegative real-valued summable functions on X. The complex case can be reduced to the real case, because the real and imaginary parts of a complex-valued function on X are summable as well. Of course, one should also verify that the value of the sum (10.1) does not depend on the particular representation of f in terms of nonnegative real-valued summable functions on X. This uses the linearity properties of the sum for nonnegative real-valued functions on X mentioned in Section 4.

Remember that the spaces ℓ^{1}(X, R), ℓ^{1}(X, C) of real and complex-valued summable functions on X, respectively, are vector spaces over R, C with respect to pointwise addition and scalar multiplication, as in Section 8. One can check that

f ↦ ∑_{x∈X} f(x) (10.2)

defines a linear functional on these spaces, using the linearity properties for sums of nonnegative real-valued functions as in Section 4 again, or the remarks in the preceding paragraph about how the sum (10.1) is well defined when f is summable on X. Moreover, we have that

|∑_{x∈X} f(x)| ≤ ∑_{x∈X} |f(x)| (10.3)

for every real or complex-valued summable function f on X. In the real case, this can be obtained by decomposing f into its positive and negative parts. In the complex case, one can get a slightly weaker estimate with an extra factor of 2 on the right side using the real case applied to the real and imaginary parts of a complex-valued summable function on X. Of course, if a complexvalued function f on X has finite support, then (10.3) reduces to the triangle inequality for the standard absolute value function on R or C. One can use this and an approximation argument to get that (10.3) holds for every complexvalued summable function on X.

It is easy to see that (10.2) is a bounded linear functional on ℓ^{1}(X, R) and ℓ^{1}(X, C), with dual norm equal to 1, using (10.3). Remember that c_{00}(X, R) and c_{00}(X, C) are dense linear subspaces of ℓ^{1}(X, R) and ℓ^{1}(X, C), respectively, as in Section 8. It follows that (10.2) is uniquely determined as a bounded linear functional on ℓ^{1}(X, R) or ℓ^{1}(X, C) by its restriction to c_{00}(X, R) or c_{00}(X, C), respectively, because bounded linear mappings are continuous. Of course, if a real or complex-valued function f on X has finite support, then (10.1) reduces to a finite sum in R or C, as appropriate.

If X = Z_{+}, then the sum (10.1) may be treated as an infinite series. In this case, the hypothesis that f be summable on Z_{+} says exactly that this infinite series is absolutely convergent. Similarly, if X = Z, then the sum may be treated as a doubly-infinite series, which reduces to a sum of two ordinary infinite series.

11 Hilbert spaces

In this section, we take k = R or C again, equipped with the standard absolute value function. Let V be a vector space over R or C, and let ⟨v, w⟩ be an inner product on V, which is Hermitian in the complex case. Thus ⟨v, v⟩ is a nonnegative real number for every v ∈ V, and we put

∥v∥ = ⟨v, v⟩^{1/2} (11.1) for each v ∈ V, using the nonnegative square root of ⟨v, v⟩ on the right side. It is well known that |⟨v, w⟩| ≤ ∥v∥ ∥w∥ (11.2)

for every v, w ∈ V, which is the Cauchy–Schwarz inequality in this context, and that ∥v∥ defines a norm on V. If V is complete with respect to the metric associated to this norm, then V is said to be a Hilbert space.

Let X be a nonempty set, and let ℓ^{2}(X, R) and ℓ^{2}(X, C) be as in Section 8. If f and g are elements of either of these spaces, then it is well known that |f(x)| |g(x)| is summable on X. This can be obtained from the Cauchy–Schwarz inequality, or by comparing this function more directly with |f(x)|^{2} + |g(x)|^{2}. The standard inner product on ℓ^{2}(X, R) is defined as usual by

⟨f, g⟩ = ∑_{x∈X} f(x) g(x), (11.3)

and similarly the standard inner product on ℓ^{2}(X, C) is defined by

⟨f, g⟩ = ∑_{x∈X} f(x) \overline{g(x)}, (11.4)

where \overline{a} denotes the complex conjugate of a complex number a. The sums on the right sides of (11.3) and (11.4) are defined as in the previous section, and it is easy to see that the norms corresponding to these inner products are the same as the ℓ^{2} norms defined in Section 8.

Now let (X, A, μ) be a measure space, and let L^{2}(X) be the corresponding space of real or complex-valued square-integrable functions on X. If f, g are square integrable functions on X, then it is well known that |f(x)| |g(x)| is also integrable on X, as before. This permits us to define

⟨f, g⟩ = ∫_{X} f(x) g(x) dμ(x) (11.5)

in the real case, and

⟨f, g⟩ = ∫_{X} f(x) \overline{g(x)} dμ(x) (11.6)

in the complex case. These determine inner products on the real and complex versions of L^{2}(X), for which the corresponding norm is the usual L^{2} norm

( ∫_{X} |f(x)|^{2} dμ(x))^{1/2}. (11.7)

Remember that square-integrable functions on X are considered the same in L^{2}(X) when they are equal almost everywhere on X with respect to μ.

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be inner product spaces, both real or both complex. A linear mapping T from V onto W is said to be unitary if

⟨T (v), T (v′)⟩_{W} = ⟨v, v′⟩_{V} (11.8) for every v, v′ ∈ V. This implies that T is an isometry with respect to the corresponding norms, by taking v = v′. Conversely, if T is an isometric linear mapping from V into W, then it is well known that T satisfies (11.8), because of polarization identities. It is easy to see directly that the shift operator T in Section 3 defines a unitary transformation from ℓ^{2}(Z, R) onto itself, and from ℓ^{2}(Z, C) onto itself.

12 Infinite series

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q-norm N for some q > 0, with respect to | · | on k. As usual, an infinite series ∑_{j=1}^{∞} v_{j} with terms in V is said to converge in V if the corresponding sequence of partial sums ∑_{j=1}^{n} v_{j} converges in V, in which case the value of the infinite series is defined to be the limit of the sequence of partial sums. Note that the sequence of partial sums is a Cauchy sequence in V if and only if for each ϵ > 0 there is a positive integer L such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (12.1)

for every n ≥ l ≥ L. In particular, this implies that {v_{j}}_{j=1}^{∞} converges to 0 in V, by taking l = n. Another necessary condition for the sequence of partial sums to be a Cauchy sequence in V is that the partial sums be bounded in V.

Suppose for the moment that q < ∞, so that

N( ∑_{j=l}^{n} v_{j})^{q} ≤ ∑_{j=l}^{n} N(v_{j})^{q} (12.2)

for every n ≥ l ≥ 1. Let us say that ∑_{j=1}^{∞} v_{j} converges q-absolutely in V if ∑_{j=1}^{∞} N(v_{j})^{q} converges as an infinite series of nonnegative real numbers. This implies that the sequence of partial sums ∑_{j=1}^{n} v_{j} is a Cauchy sequence in V, using the reformulation of the Cauchy condition mentioned in the previous paragraph. If V is complete with respect to the q-metric associated to N, then it follows that ∑_{j=1}^{∞} v_{j} converges in V. Under these conditions, one can also check that

N( ∑_{j=1}^{∞} v_{j})^{q} ≤ ∑_{j=1}^{∞} N(v_{j})^{q}, (12.3)

using (12.2) with l = 1.

Suppose now that q = ∞, so that

N( ∑_{j=l}^{n} v_{j}) ≤ max_{l≤j≤n} N(v_{j}) (12.4)

for every n ≥ l ≥ 1. If {v_{j}}_{j=1}^{∞} converges to 0 in V, then it is easy to see that the sequence of partial sums ∑_{j=1}^{n} v_{j} forms a Cauchy sequence, using the earlier reformulation of the Cauchy condition again. If V is also complete with respect to the ultrametric associated to N, then ∑_{j=1}^{∞} v_{j} converges in V, and we get that

N( ∑_{j=1}^{∞} v_{j}) ≤ sup_{j≥1} N(v_{j}), (12.5)

using (12.4) with l = 1. Note that the supremum on the right side of (12.5) is attained in this situation, because N(v_{j}) → 0 as j → ∞, by hypothesis.

Let us now take k = R or C, with the standard absolute value function, and let (V, ⟨v, w⟩) be a real or complex inner product space, with the corresponding norm ∥v∥. Suppose that ∑_{j=1}^{∞} v_{j} is an infinite series of pairwise-orthogonal vectors in V, so that ⟨v_{j}, v_{l}⟩ = 0 (12.6)

when j ≠ l. This implies that

∥∑_{j=l}^{n} v_{j}∥^{2} = ∑_{j=l}^{n} ∥v_{j}∥^{2} (12.7)

for each n ≥ l ≥ 1. If ∑_{j=1}^{∞} ∥v_{j}∥^{2} converges as an infinite series of nonnegative real numbers, then it follows that the sequence of partial sums ∑_{j=1}^{n} v_{j} is a Cauchy sequence in V. If V is a Hilbert space, then ∑_{j=1}^{∞} v_{j} converges in V, and

∥∑_{j=1}^{∞} v_{j}∥^{2} = ∑_{j=1}^{∞} ∥v_{j}∥^{2}. (12.8)

Of course, ∑_{j=1}^{∞} ∥v_{j}∥^{2} converges if and only if the partial sums ∑_{j=1}^{n} ∥v_{j}∥^{2} are bounded. In this situation, this happens exactly when the partial sums ∑_{j=1}^{n} v_{j} are bounded in V.

13 Completeness

Let (X, d_{X}) and (Y, d_{Y}) be q_{X}, q_{Y} -metric spaces for some q_{X}, q_{Y} > 0, and let E be a dense subset of X. Also let f be a uniformly continuous mapping from E into Y, with respect to the restriction of d_{X} to elements of E. If Y is complete with respect to d_{Y}, then there is a unique extension of f to a uniformly continuous mapping from X into Y. This is well known for ordinary metric spaces, and essentially the same argument works for q-metric spaces. One can also reduce to the case of ordinary metric spaces, by replacing d_{X} or d_{Y} with d_{X}^{q_{X}} or d_{Y}^{q_{Y}} when q_{X} ≤ 1 or q_{Y} ≤ 1, respectively. More precisely, uniform continuity of f is needed for the existence of such an extension, but ordinary continuity of the extension is sufficient for its uniqueness. Similarly, completeness of Y is only used for the existence of the extension, and not uniqueness.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k with q_{V}, q_{W} -norms N_{V}, N_{W}, respectively, for some q_{V} , q_{W} > 0, and with respect to | · | on k. Also let V_{0} be a dense linear subspace of V, and let T be a bounded linear mapping from V_{0} into W, with respect to the restriction of N_{V} to V_{0}. Thus T is uniformly continuous with respect to the q_{V}, q_{W} -metrics associated to N_{V}, N_{W}, respectively, as in Section 9. If W is complete with respect to the q_{W} -metric associated to N_{W}, then T has a unique extension to a uniformly continuous mapping from V into W, as in the preceding paragraph. In this context, one can check that this extension is a bounded linear mapping from V into W, with the same operator norm as T has on V_{0}.

Let X be a nonempty set, and let us take k = R or C with the standard absolute value function for the moment. If f is a real or complex-valued function on X with finite support, then one can define ∑_{x∈X} f(x) as a real or complex number by reducing to a finite sum. It is easy to see that

|∑_{x∈X} f(x)| ≤ ∑_{x∈X} |f(x)| (13.1)

under these conditions, using the ordinary triangle inequality on R or C. Of course,

f ↦ ∑_{x∈X} f(x) (13.2)

defines linear functionals on c_{00}(X, R) and c_{00}(X, C). More precisely, these are bounded linear functionals on c_{00}(X, R) and c_{00}(X, C) with respect to the restrictions of the corresponding ℓ^{1} norms to these spaces, with dual norms equal to 1, because of (13.1). Remember that c_{00}(X, R) and c_{00}(X, C) are dense in ℓ^{1}(X, R) and ℓ^{1}(X, C), with respect to the metrics corresponding to the ℓ^{1} norms, as in Section 8. It follows that there are unique extensions of (13.2) to bounded linear functionals on ℓ^{1}(X, R) and ℓ^{1}(X, C) with dual norms equal to 1, as in the preceding paragraph, and because R and C are complete with respect to their standard metrics. These extensions can also be obtained as in Section 10.

Now let k be any field with an ultrametric absolute value function | · |. If f is a k-valued function on X with finite support, then ∑_{x∈X} f(x) can be defined as an element of k, by reducing to a finite sum, as before. In this case, we have that

|∑_{x∈X} f(x)| ≤ max_{x∈X} |f(x)|, (13.3)

by the ultrametric version of the triangle inequality. This implies that (13.2) defines a bounded linear functional on c_{00}(X, k) with respect to the supremum ultranorm, with dual norm equal to 1. As in Section 8, c_{00}(X, k) is dense in c_{0}(X, k) with respect to the ultrametric associated to the supremum ultranorm. If k is complete with respect to the ultrametric associated to | · |, then there is a unique extension of (13.2) to a bounded linear functional on c_{0}(X, k) with respect to the supremum ultranorm, and with dual norm equal to 1. This extension may be used as the definition of ∑_{x∈X} f(x) when f is a k-valued function on X that vanishes at infinity and k is complete. If X = Z_{+}, then this sum can be treated as an infinite series, as in the previous section. Similarly, if X = Z, then the sum can be reduced to a sum of two infinite series.

14 Orthonormal systems

In this section, we take k = R or C again, with the standard absolute value function. Let (V, ⟨v, w⟩_{V}) be a real or complex inner product space, and let ∥v∥_{V} be the corresponding norm on V. Suppose for the moment that v_{1}, …, v_{n} are finitely many orthonormal vectors in V, so that ∥v_{j}∥_{V} = 1 for each j = 1, …, n, and ⟨v_{j}, v_{l}⟩_{V} = 0 (14.1)

when j ≠ l. Let v ∈ V be given, and put

w = ∑_{j=1}^{n}⟨v, v_{j}⟩_{V} v_{j}. (14.2)

By construction, ⟨v, v_{j}⟩_{V} = ⟨w, v_{j}⟩_{V} for each j = 1, …, n, so that v − w is orthogonal to v_{j}. This implies that v − w is orthogonal to w, and hence

(14.3)  ∥v∥_{V}^{2} = ∥(v − w) + w∥_{V}^{2} = ∥v − w∥_{V}^{2} + ∥w∥_{V}^{2}
= ∥v − w∥_{V}^{2} + ∑_{j=1}^{n} |⟨v, v_{j}⟩_{V} |^{2}.

In particular, we have that

∑_{j=1}^{n} |⟨v, v_{j}⟩_{V} |^{2} ≤ ∥v∥_{V}^{2} . (14.4)

If v is in the linear span of v_{1}, …, v_{n} in V, then w = v, and equality holds in (14.4). If u is any element of the linear span of v_{1}, …, v_{n} in V, then w − u is also in the linear span of v_{1}, …, v_{n}, so that v − w is orthogonal to w − u. This implies that

(14.5) ∥v − u∥_{V}^{2} = ∥(v − w) + (w − u)∥_{V}^{2} = ∥v − w∥_{V}^{2} + ∥w − u∥_{V}^{2}
≥ ∥v − w∥_{V}^{2} ,

and hence that w minimizes the distance to v in V among elements of the linear span of v_{1}, …, v_{n}.

Now let A be a nonempty set, and let {v_{a}}_{a∈A} be an orthonormal family of vectors in V indexed by A. As before, this means that ∥v_{a}∥_{V} = 1 for every a ∈ A, and that ⟨v_{a}, v_{b}⟩_{V} = 0 when a ≠ b. If v ∈ V, then

∑_{a∈A} |⟨v, v_{a}⟩_{V} |^{2} ≤ ∥v∥_{V}^{2} . (14.6)

More precisely, the sum on the left is defined as in Section 4, as the supremum of the corresponding sums over all finite subsets of A. These finite subsums can be estimated as in (14.4), to get (14.6). If v is an element of the linear span of the v_{a}’s, a ∈ A, in V, then equality holds in (14.6), as before. Similarly, if v is an element of the closure of the linear span of the v_{a}’s, a ∈ A, in V, then one can check that equality holds in (14.6).

Let c_{00}(A) and ℓ^{2}(A) denote c_{00}(A, k) and ℓ^{2}(A, k) with k = R or C, as appropriate. If f ∈ c_{00}(A), then

T (f) = ∑_{a∈A} f(a) v_{a} (14.7)

reduces to a finite sum in V, so that T defines a linear mapping from c_{00}(A) into V. It is easy to see that

⟨T (f), T (g)⟩_{V} = ⟨f, g⟩_{ℓ^{2}(A)} (14.8)

for every f, g ∈ c_{00}(A), where the right side of (14.8) is given by (11.3) or (11.4), as appropriate, with X = A. In particular, we have that

∥T (f)∥_{V} = ∥f∥_{ℓ^{2}(A)} (14.9)

for every f ∈ c_{00}(A), by taking f = g in (14.8). Note that T maps c_{00}(A) onto the linear span of the v_{a}’s, a ∈ A, in V.

If V is a Hilbert space, then T has a unique extension to a bounded linear mapping from ℓ^{2}(A) into V, as in the previous section. This extension satisfies (14.8) for every f, g ∈ ℓ^{2}(A), and hence (14.9) for every f ∈ ℓ^{2}(A). One can check that this extension maps ℓ^{2}(A) onto the closure of the linear span of the v_{a}’s, a ∈ A, in V. If A = Z_{+}, then the right side of (14.7) can be treated as an infinite series in V, as in Section 12. Similarly, if A = Z, then (14.7) can be reduced to a sum of two infinite series in V.

15 Fourier series

In this section, we take k = C, with the standard absolute value function. Let

T = {z ∈ C : |z| = 1} (15.1)

be the unit circle in C, and let μ be a complex Borel measure on T. If j ∈ Z, then the jth Fourier coefficient of μ is defined by

μ̂(j) = ∫_{T} \overline{z}^{j} dμ(z). (15.2)

Observe that |μ̂(j)| ≤ |μ|(T) (15.3) for each j ∈ Z, where |μ| denotes the total variation measure on T associated to μ. It is well known that μ is uniquely determined by its Fourier coefficients, in the following way.

Let C(T) be the space of complex-valued continuous functions on T, which is a vector space over C with respect to pointwise addition and scalar multiplication. Observe that λ_{μ}(ϕ) = ∫_{T} ϕ dμ (15.4)

defines a linear functional on C(T), and that

|λ_{μ}(ϕ)| ≤ ( sup_{z∈T} |ϕ(z)|) |μ|(T) (15.5)

for every ϕ ∈ C(T). Thus λ_{μ} is a bounded linear functional on C(T) with respect to the supremum norm on C(T), and the corresponding dual norm of λ_{μ} is less than or equal to |μ|(T). It is well known that μ automatically satisfies certain regularity properties in this situation, which implies that μ is uniquely determined by λ_{μ} on C(T). In fact, |μ|(T) is equal to the dual norm of λ_{μ} with respect to the supremum norm on C(T). It is also well known that every bounded linear functional on C(T) with respect to the supremum norm corresponds to a complex Borel measure on T in this way. In order to show that μ is uniquely determined by its Fourier coefficients, it suffices to show that λ_{μ} is uniquely determined as a linear functional on C(T) by the Fourier coefficients of μ.

Note that (15.2) is the same as λ_{μ} applied to \overline{z}^{j} as a complex-valued continuous function on T for each j ∈ Z. If μ̂(j) = 0 for each j ∈ Z, then it follows that λ_{μ}(ϕ) = 0 (15.6) when ϕ is of the form \overline{z}^{j} for some j ∈ Z. This implies that (15.6) also holds when ϕ is a linear combination of finitely many such functions, by linearity. It is well known that these linear combinations are dense in C(T) with respect to the supremum norm. Under these conditions, we get that (15.6) holds for every ϕ ∈ C(T), because λ_{μ} is bounded on C(T) with respect to the supremum norm, as desired.

Let ν be the complex Borel measure on T defined by putting

ν(E) = ∫_{E} z dμ(z) (15.7)

for every Borel set E ⊆ T. It is well known that

∫_{T} ϕ(z) dν(z) = ∫_{T} ϕ(z) z dμ(z) (15.8)

for every bounded complex-valued Borel measurable function ϕ on T. In particular, for each j ∈ Z, we have that

ν̂(j) = ∫_{T} \overline{z}^{j} dν(z) = ∫_{T} \overline{z}^{j−1} dμ(z) = μ̂(j − 1). (15.9)

Note that |ν| = |μ| as nonnegative Borel measures on T.

16 Fourier series, continued

Let us continue to take k = C with the standard absolute value function. Of course, there is an analogue of Lebesgue measure on the unit circle T, which corresponds to ordinary Lebesgue measure on [0, 2π) with respect to an arclength parameterization of T. If f is a complex-valued integrable function on T with respect to this measure, then the jth Fourier coefficient of f is defined for each j ∈ Z by

f̂(j) = \frac{1}{2π} ∫_{T} f(z) \overline{z}^{j} |dz|, (16.1)

where |dz| denotes the arclength measure on T just mentioned. In this case, we have that

|f̂(j)| ≤ \frac{1}{2π} ∫_{T} |f(z)| |dz| (16.2)

for every j ∈ Z, and

lim_{|j|→∞} f̂(j) = 0, (16.3)

by the Riemann–Lebesgue lemma. If we put

f_{1}(z) = z f(z) (16.4)

for each z ∈ T, then f_{1} is integrable on T too, and

f̂_{1}(j) = \frac{1}{2π} ∫_{T} f(z) \overline{z}^{j−1} = f̂(j − 1) (16.5)

for every j ∈ Z.

In this situation,

μ_{f}(E) = \frac{1}{2π} ∫_{E} f(z) |dz| (16.6)

defines a complex Borel measure on T, with

|μ_{f}|(E) = \frac{1}{2π} ∫_{E} |f(z)| |dz| (16.7)

for every Borel set E ⊆ T. If ϕ is a bounded complex-valued Borel measurable function on T, then

∫_{T} ϕ dμ_{f} = ∫_{T} ϕ(z) f(z) |dz|. (16.8)

In particular, μ̂_{f}(j) = f̂(j) (16.9)

for each j ∈ Z, where μ̂_{f} is as defined in (15.2). Similarly, (16.2) corresponds to (15.3) with μ = μ_{f}, μ_{f_{1}} corresponds to (15.7), and (16.5) corresponds to (15.9). Note that f is uniquely determined by its Fourier coefficients, as in the previous section.

Let L^{2}(T) be the space of complex-valued square-integrable functions on T with respect to arclength measure. This is a Hilbert space with respect to the inner product

⟨f, g⟩ = ⟨f, g⟩_{L^{2}(T)} = \frac{1}{2π} ∫_{T} f(z) \overline{g(z)} |dz|, (16.10)

and the corresponding norm

∥f∥_{2} = ∥f∥_{L^{2}(T)} = (\frac{1}{2π} ∫_{T} |f(z)|^{2} |dz|)^{1/2}. (16.11)

It is well known that ∫_{T} z^{j} |dz| = 0 (16.12)

for every j ∈ Z with j ≠ 0, which implies that the functions z^{j} on T with j ∈ Z are orthonormal with respect to (16.10). If f ∈ L^{2}(T), then f̂(j) is the same as the inner product of f with z^{j} with respect to (16.10) for each j ∈ Z. This implies that

∑_{j=−∞}^{∞} |f̂(j)|^{2} ≤ \frac{1}{2π} ∫_{T} |f(z)|^{2} |dz|, (16.13)

as in (14.6). It is well known that the linear span of the functions z^{j} with j ∈ Z is dense in L^{2}(T), since it is dense in C(T) with respect to the supremum norm. Thus we actually have that

∑_{j=−∞}^{∞} |f̂(j)|^{2} = \frac{1}{2 π} ∫_{T} |f(z)|^{2} (16.14)

for every f ∈ L^{2}(T), as in Section 14.

Let {a_{j}}_{j=−∞}^{∞} be a doubly-infinite sequence of complex numbers that is square-summable, in the sense that

∑_{j=−∞}^{∞} |a_{j}|^{2} < ∞. (16.15)

Thus {a_{j}}_{j=−∞}^{∞} corresponds to an element of ℓ^{2}(Z, C), and

f(z) = ∑_{j=−∞}^{∞} a_{j} z^{j} (16.16)

determines an element of L^{2}(T), because of the orthonormality of the functions z^{j} in L^{2}(T), as in Sections 12 and 14. By construction,

f̂(j) = a_{j} (16.17)

for each j ∈ Z, and

∑_{j=−∞}^{∞} a_{j−1} z^{j} = ∑_{j=−∞}^{∞} a_{j} z^{j+1} = z f(z). (16.18)

If {a_{j}}_{j=−∞}^{∞} is summable, in the sense that

∑_{j=−∞}^{∞} |a_{j}| < ∞, (16.19)

then the right side of (16.16) can be treated as the sum of a summable complexvalued function on Z for each z ∈ T, as in Section 10. In this case, the right side of (16.16) can also be considered as the sum of two absolute convergent series in C(T) with respect to the supremum norm, as in Section 12.

17 Some bounded linear functionals

Let us begin with the case where k = R or C, with the standard absolute value function. Let (V, ⟨v, w⟩) be a real or complex inner product space. If w ∈ V, then λ_{w}(v) = ⟨v, w⟩ (17.1)

defines a linear functional on V. This linear functional is bounded on V with respect to the norm ∥v∥ corresponding to the inner product, because

|λ_{w}(v)| ≤ ∥v∥ ∥w∥ (17.2)

for every v ∈ V, by the Cauchy–Schwarz inequality. More precisely, this implies that the corresponding dual norm of λ_{w} is less than or equal to ∥w∥. It is easy to see that the dual norm of λ_{w} is equal to ∥w∥, since

λ_{w}(w) = ⟨w, w⟩ = ∥w∥^{2}. (17.3)

If V is a Hilbert space, then it is well known that every bounded linear functional on V is of this form. Note that this representation is unique, because λ_{w} ≠ 0 when w ≠ 0.

Let X be a nonempty set, and let us use the notation c_{00}(X), c_{0}(X), and ℓ^{r}(X) for the usual spaces of real or complex-valued functions on X, as appropriate. Suppose that 1 ≤ r, r′ ≤ ∞ are conjugate exponents, in the sense that 1/r + 1/r′ = 1. (17.4)

If f ∈ ℓ^{r}(X) and g ∈ ℓ^{r′}(X), then Hölder’s inequality implies that their product f(x) g(x) is summable on X, with

∑_{x∈X} |f(x)| |g(x)| ≤ ∥f∥_{r} ∥g∥_{r′}. (17.5)

Thus

λ_{g}(f) = ∑_{x∈X} f(x) g(x) (17.6)

can be defined as a real or complex number, as appropriate, as in Section 10. We also have that |λ_{g}(f)| ≤ ∥f∥_{r} ∥g∥_{r′}, (17.7) by (17.5). This implies that λ_{g} defines a bounded linear functional on ℓ^{r}(X) for each g in ℓ^{r′}(X), with dual norm less than or equal to ∥g∥_{r′}. One can check that the dual norm of λ_{g} on ℓ^{r}(X) is equal to g, by considering suitable f ∈ ℓ^{r}(X).

If 1 ≤ r < ∞, then it is well known that every bounded linear functional on ℓ^{r}(X) is of this form. Remember that any linear functional on c_{00}(X) can be expressed as (17.6) for some real or complex-valued function g on X, as apprpriate, and as in Section 2. If this linear functional is bounded on c_{00}(X) with respect to ∥f∥_{r}, then one can verify that g ∈ ℓ^{r′}(X). More precisely, one can show that ∥g∥_{r′} is less than the corresponding dual norm of λ_{g} on c_{00}(X) with respect to ∥f∥_{r} on c_{00}(X), by considering suitable f ∈ c_{00}(X), as in the preceding paragraph. It follows that λ_{g} defines a bounded linear functional on ℓ^{r}(X), as in the previous paragraph again. If 1 ≤ r < ∞, then c_{00}(X) is dense in ℓ^{r}(X), and the original bounded linear functional on ℓ^{r}(X) is equal to λ_{g} on all of ℓ^{r}(X). Essentially the same argument shows that a bounded linear functional on c_{0}(X) with respect to the supremum norm can be expressed as (17.6) for some g ∈ ℓ^{1}(X). Note that the case where r = r′ = 2 also corresponds to the earlier discussion of Hilbert spaces.

Suppose now that g ∈ ℓ^{∞}(X), so that λ_{g} as in (17.6) defines a bounded linear functional on ℓ^{1}(X). In this case, (17.5) is very simple, and one can get that the dual norm of λ_{g} on ℓ^{1}(X) is equal to ∥g∥_{∞} using (2.6). If 0 < r ≤ 1, then ℓ^{r}(X) is contained in ℓ^{1}(X), and

∥f∥_{1} ≤ ∥f∥_{r} (17.8)

for every f ∈ ℓ^{r}(X), as in (8.4) and (8.5). This implies that the restriction of λ_{g} to ℓ^{r}(X) defines a bounded linear functional on ℓ^{r}(X), with dual norm less than or equal to ∥g∥_{∞}. One can check that the dual norm of λ_{g} on ℓ^{r}(X) is equal to ∥g∥_{∞} when 0 < r ≤ 1, using (2.6) again.

It is well known that every bounded linear functional on ℓ^{r}(X) is of this form when 0 < r ≤ 1. As before, the restriction of a bounded linear functional on ℓ^{r}(X) to c_{00}(X) can be expressed as in (17.6) for some real or complex-valued function g on X. The boundedness of this linear functional on c_{00}(X) with respect to ∥f∥_{r} implies that g ∈ ℓ^{∞}(X), because of (2.6). This implies that λ_{g} extends to a bounded linear functional on ℓ^{r}(X) when 0 < r ≤ 1, as in the preceding paragraph. The original bounded linear functional on ℓ^{r}(X) is equal to λ_{g} on ℓ^{r}(X), because c_{00}(X) is dense in ℓ^{r}(X) when r < ∞.

Now let k be a field with an ultrametric absolute value function | · |, and suppose that k is complete with respect to the associated ultrametric. If f, g are bounded k-valued functions on X and at least one of f, g vanishes at infinity on X, then it is easy to see that their product f(x) g(x) vanishes at infinity on X too. This permits us to define λ_{g}(f) as an element of k as in (17.6), as discussed in Section 13. We also get that

|λ_{g}(f)| ≤ max_{x∈X}(|f(x)| |g(x)|) ≤ ∥f∥_{∞} ∥g∥_{∞} (17.9)

where the first step is as in Section 13. This implies that λ_{g} defines a bounded linear functional on c_{0}(X, k) with respect to the supremum ultranorm when g is in ℓ^{∞}(X, k), and similarly that λ_{g} defines a bounded linear functional on ℓ^{∞}(X, k) when g ∈ c_{0}(X, k). In both cases, (17.9) implies that the corresponding dual norm of λ_{g} is less than or equal to ∥g∥_{∞}, and one can check that the dual norm of λ_{g} is equal to ∥g∥_{∞}, using (2.6). Of course, if X has only finitely many elements, then the completeness of k is not needed, and vanishing at infinity on X is trivial.

It is easy to see that every bounded linear functional on c_{0}(X, k) is of this form, using the same type of argument as in the archimedian case. More precisely, every linear functional on c_{00}(X, k) can be expressed as in (17.6) for some k-valued function g on X, as in Section 2. If this linear functional on c_{00}(X, k) is bounded with respect to the supremum ultranorm, then one can verify that g is bounded on X, using (2.6) again. This implies that λ_{g} defines a bounded linear functional on c_{0}(X, k) with respect to the supremum ultranorm, as in the preceding paragraph. The original bounded linear functional on c_{0}(X, k) is equal to λ_{g} on all of c_{0}(X, k), because c_{00}(X, k) is dense in c_{0}(X, k) with respect to the supremum norm.

Let g be any bounded k-valued function on X again, so that (17.6) defines a bounded linear functional λ_{g} on c_{0}(X, k), as before. If r is a positive real number, then ℓ^{r}(X, k) is contained in c_{0}(X, k), as in (8.11). We also have that

∥f∥_{∞} ≤ ∥f∥_{r} (17.10)

for every f ∈ ℓ^{r}(X, k), as in (8.4). Hence the restriction of λ_{g} to ℓ^{r}(X, k) defines a bounded linear functional on ℓ^{r}(X, k), with dual norm less than or equal to ∥g∥_{∞}. As before, one can verify that the dual norm of λ_{g} on ℓ^{r}(X, k) is equal to ∥g∥_{∞}, using (2.6).

Every bounded linear functional on ℓ^{r}(X, k) is of this form when 0 < r < ∞, by an argument like the one in the archimedian case. More precisely, any linear functional on c_{00}(X, k) can be expressed as in (17.6) for some k-valued function g on X, as before. If this linear functional on c_{00}(X, k) is bounded with respect to ∥f∥_{r}, then g is bounded on X, because of (2.6). Thus λ_{g} extends to a bounded linear functional on ℓ^{r}(X, k) when 0 < r < ∞, as in the preceding paragraph. It follows that the original linear functional on ℓ^{r}(X, k) is equal to λ_{g} on all of ℓ^{r}(X, k), because c_{00}(X, k) is dense in ℓ^{r}(X, k) when r < ∞.

18 Some bounded linear mappings

Let k be a field, let X be a nonempty set, and let V be a vector space over k. Also let a(x) be a V -valued function on X, and put

T_{a}(f) = ∑_{x∈X} f(x) a(x) (18.1)

for every f ∈ c_{00}(X, k). More precisely, the right side reduces to a finite sum in V, and so defines an element of V. Thus T_{a} defines a linear mapping from c_{00}(X, k) into V. Note that T_{a}(δ_{y}) = a(y) (18.2)

for every y ∈ X, where δ_{y} ∈ c_{00}(X, k) is as in (2.2). It is easy to see that every linear mapping from c_{00}(X, k) into V is of this form, since the δ_{y}’s with y ∈ X form a basis of c_{00}(X, k) as a vector space over k. More precisely, a linear mapping T from c_{00}(X, k) into V can be expressed as T_{a}, where a(y) = T (δ_{y}) for each y ∈ X.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N_{V} be a q_{V} -norm on V with respect to | · | on k for some q_{V} > 0. Also let a be a V -valued function on X again, so that (18.1) defines a linear mapping from c_{00}(X, k) into V, and let 0 < r ≤ ∞ be given. If T_{a} is bounded with respect to ∥f∥_{r} on c_{00}(X, k) and N_{V} on V, then it is easy to see that N_{V} (a(y)) is less than or equal to the corresponding operator q-norm of V for every y ∈ X, because of (18.2). In the other direction, if N_{V} (a(x)) is bounded on X, and if r ≤ q_{V}, then we have that

N_{V} (T_{a}(f)) ≤ ( sup_{x∈X} N_{V} (a(x))) ∥f∥_{r} (18.3)

for every f ∈ c_{00}(X, k). To see this, remember that N_{V} is an r-norm on V when r ≤ q_{V}, as in Section 7. Using this, (18.3) follows easily from the definition of T_{a} and the r-norm version of the triangle inequality. Of course, (18.3) says that T_{a} is bounded with respect to ∥f∥_{r} on c_{00}(X, k) and N_{V} on V, with operator norm less than or equal to sup_{x∈X} N_{V} (a(x)). In this situation, the operator norm of T_{a} is equal to this supremum, since the reverse inequality holds for every r > 0, as before.

Suppose for the moment that V is complete with respect to the q_{V} -metric associated to N_{V}. Let us continue to ask that N_{V} (a(x)) be bounded on X, and that 0 < r ≤ q_{V}. If r < ∞, then c_{00}(X, k) is dense in ℓ^{r}(X, k), and it follows that T_{a} extends to a bounded linear mapping from ℓ^{r}(X, k) into V, as in Section 13. Similarly, if r = ∞, then c_{00}(X, k) is dense in c_{0}(X, k) with respect to ∥f∥_{∞}, and T_{a} extends to a bounded linear mapping from c_{0}(X, k) into V, using ∥f∥_{∞} on c_{0}(X, k). In these cases, the sum on the right side of (18.1) can be treated as an infinite series when X = Z_{+}, or reduced to a sum of two infinite series when X = Z.

If λ_{x} is a linear functional on V for each x ∈ X, then

v ↦ λ_{x}(v) (18.4)

defines a linear mapping from V into the space c(X, k) of all k-valued functions on X. Clearly every linear mapping from V into c(X, k) corresponds to a family of linear functionals on V indexed by X in this way. If λ_{x} is a bounded linear functional on V with respect to N_{V} for each x ∈ X, and if the corresponding dual norm of λ_{x} is uniformly bounded as a function of x ∈ X, then (18.4) defines a bounded linear mapping from V into ℓ^{∞}(X, k). More precisely, the operator q_{k}-norm of this linear mapping from V into ℓ^{∞}(X, k) is equal to the supremum of the dual norm of λ_{x} on V with respect to N_{V} over x ∈ X. As before, every bounded linear mapping from V into ℓ^{∞}(X, k) corresponds to a family of bounded linear functionals on V indexed by X with uniformly bounded dual norms.

19 Multiplication operators

Let k be a field, and let X be a nonempty set. The space c(X, k) of k-valued functions on X is a commutative algebra over k, with respect to pointwise multiplication of functions. The constant function 1_{X} = 1_{X,k} on X equal to the multiplicative identity element 1 = 1_{k} in k at every point is the multiplicative identity element in c(X, k). A k-valued function a has a multiplicative inverse in c(X, k) if and only if a(x) ≠ 0 for every x ∈ X, in which case the multiplicative inverse is given by 1/a(x). The space c_{00}(X, k) of k-valued functions on X with finite support is an ideal in c(X, k).

Put M_{a}(f) = a f (19.1)

for all k-valued functions a, f on X. This defines a linear mapping M_{a} from c(X, k) into itself, which is the multiplication operator associated to a. Note that M_{a} sends c_{00}(X, k) into itself. Clearly

M_{a}(1_{X}) = a (19.2)

for every a ∈ c(X, k). If y ∈ X and δ_{y} ∈ c_{00}(X, k) is as in (2.2), then

M_{a}(δ_{y}) = a(y) δ_{y} (19.3)

for every a ∈ c(X, k). The mapping

a ↦ M_{a} (19.4)

is an algebra homomorphism from c(X, k) into the algebra of linear mappings from c(X, k) into itself. In particular, this mapping sends a = 1_{X} to the identity operator on c(X, k).

Let | · | be a q-absolute value function on k for some q > 0, so that ℓ^{r}(X, k) can be defined for each r > 0 as in Section 8. If a is a bounded k-valued function on X, then M_{a} maps ℓ^{r}(X, k) into itself for every r > 0. More precisely, the restriction of M_{a} to ℓ^{r}(X, k) is a bounded linear mapping with respect to the usual ℓ^{r} norm, and the corresponding operator norm of M_{a} is equal to the supremum norm ∥a∥_{∞} of a. Similarly, M_{a} maps the space c_{0}(X, k) of k-valued functions on X that vanish at infinity into itself when a is bounded on X. In this case, the operator norm of M_{a} with respect to the restriction of the supremum norm to c_{0}(X, k) is equal to ∥a∥_{∞} too.

Let us now take k = R or C, with the standard absolute value function. Let (X, A, μ) be a measure space, so that X is a set, A is a σ-algebra of measurable subsets of X, and μ is a nonnegative measure defined on A. The corresponding Lebesgue spaces L^{r}(X) can be defined in the usual way for each r > 0. Remember that the L^{r} norm defines a norm on L^{r}(X) in the usual sense when r ≥ 1, and it defines an r-norm on L^{r}(X) when 0 < r ≤ 1. If a is an essentially bounded measurable real or complex-valued function on X, as appropriate, then the corresponding multiplication operator M_{a} defines a bounded linear mapping from L^{r}(X) into itself for each r > 0. The operator norm of M_{a} on L^{r}(X) is less than or equal to the essential supremum norm ∥a∥_{∞} = ∥a∥_{L^{∞}(X)} of a on X for each r > 0. It is easy to see that the operator norm of M_{a} on L^{∞}(X) is equal to ∥a∥_{∞}. In order to get that the operator norm of M_{a} on L^{r}(X) is equal to ∥a∥_{∞} when r < ∞, an additional condition on X is needed. Namely, this works when every measurable set A ⊆ X with μ(A) > 0 has a measurable subset B such that 0 < μ(B) < ∞.

20 Dual linear mappings

Let k be a field with a q_{k}-absolute value function for some q_{k} > 0, let V be a vector space over k, and let N_{V} be a q_{V} -norm N_{V} on V with respect to | · | on k for some q_{V} > 0. The dual space V′ associated to V is the space of bounded linear functionals on V, as in Section 9. This is the same as BL(V, k), where k is considered as a one-dimensional vector space over itself, and | · | is considered as a q_{k}-norm on k as a vector space. In particular, V′ is a vector space over k, which may be considered as a linear subspace of the algebraic dual V^{alg}. The corresponding operator q_{k}-norm on V′ is known as the dual q_{k}-norm, as before, and is denoted N_{V′}.

Let W be another vector space over k, and let N_{W} be a q_{W} -norm on W with respect to | · | on k for some q_{W} > 0. Also let T be a bounded linear mapping from V into W with respect to N_{V} and N_{W}, respectively. If λ is a bounded linear functional on W, then

T′(λ) = λ ∘ T (20.1)

defines a bounded linear functional on V. More precisely,

N_{V′}(T′(λ)) ≤ ∥T ∥_{op,V W} N_{W′}(λ), (20.2)

as in (9.13), where ∥T ∥_{op,V W} is the corresponding operator q_{W} -norm of T, as in (9.3). It is easy to see that T′ defines a linear mapping from W′ into V′, which is the same as the restriction of the algebraic dual linear mapping T^{alg} to V′. Using (20.2), we get that T′ is bounded with respect to the corresponding dual norms, with ∥T′∥_{op,W′V′} ≤ ∥T ∥_{op,V W} . (20.3)

Observe that T ↦ T′ (20.4)

is a linear mapping from BL(V, W) into BL(W′, V′), as in (1.2). This linear mapping is bounded with respect to the corresponding operator norms, by (20.3).

Suppose for the moment that V = W, and that N_{V} = N_{W}. As in Section 9, the identity mapping I_{V} is a bounded linear mapping from V into itself. The corresponding dual mapping is the identity mapping I_{V′} on V′, so that

(I_{V})′ = I_{V′}, (20.5)

as in (1.3).

Let Z be a third vector space over k, with a q_{Z}-norm N_{Z} with respect to | · | on k for some q_{Z} > 0. Suppose that T_{1} and T_{2} are bounded linear mappings from V into W and from W into Z, respectively, so that T_{2} ∘ T_{1} is a bounded linear mapping from V into Z. Under these conditions, we have that

(T_{2} ∘ T_{1})′ = T′_{1} ∘ T′_{2} (20.6)

as linear mappings from Z′ into V′, as in (1.5). Suppose now that T is a oneto-one bounded linear mapping from V onto W whose inverse T^{−1} is bounded as a linear mapping from W onto V. In this case, one can check that T′ is a one-to-one linear mapping from W′ onto V′. The inverse of T′ is given by

(T′)^{−1} = (T^{−1})′, (20.7)

as linear mappings from V′ into W′, as in (1.3). In particular, the inverse of T′ is bounded as a linear mapping from V′ into W′, with respect to the corresponding dual norms.

21 Hilbert space adjoints

In this section, k = R or C, with the standard absolute value function. Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be inner product spaces, both real or both complex, with corresponding norms ∥ · ∥_{V} and ∥ · ∥_{W}, respectively. If u ∈ W, then

∥u∥_{W} = sup{|⟨u, w⟩_{W} | : w ∈ W, ∥w∥_{W} ≤ 1}. (21.1)

More precisely, the right side of (21.1) is less than or equal to ∥u∥_{W}, by the Cauchy–Schwarz inequality. The opposite inequality is trivial when u = 0, and otherwise can be obtained by taking w = u/∥u∥_{W}.

Let T be a linear mapping from V into W, and consider the following condition on T:

(21.2) |⟨T (v), w⟩_{W} | is bounded for v ∈ V and w ∈ W with ∥v∥_{V} , ∥w∥_{W} ≤ 1.

Of course, this means that

sup{|⟨T (v), w⟩_{W} | : v ∈ V, w ∈ W, ∥v∥_{V} , ∥w∥_{W} ≤ 1} (21.3)

is defined as a nonnegative real number. Observe that

∥T (v)∥_{W} = sup{|⟨T (v), w⟩_{W} | : w ∈ W, ∥w∥_{W} ≤ 1} (21.4) for every v ∈ V, by applying (21.1) to u = T (v). Thus (21.2) is equivalent to following condition on T:

(21.5) ∥T (v)∥_{W} is bounded for v ∈ V with ∥v∥_{V} ≤ 1.

In this case, (21.3) is equal to

sup{∥T (v)∥_{W} : v ∈ V, ∥v∥_{V} ≤ 1}, (21.6)

because of (21.4). Note that (21.5) corresponds to (9.8) in this situation, and that (21.6) corresponds to (9.9). Thus T is bounded as a linear mapping from V into W if and only if (21.5) holds, as in Section 9, in which case the operator norm ∥T ∥_{op,V W} of T is equal to (21.6). It follows that (21.2) holds if and only if T is bounded as a linear mapping from V into W, in which case ∥T ∥_{op,V W} is equal to (21.3).

Let T be a bounded linear mapping from V into W, and observe that

|⟨T (v), w⟩_{W} | ≤ ∥T (v)∥_{W} ∥w∥_{W} ≤ ∥T ∥_{op,V W} ∥v∥_{V} ∥w∥_{W} (21.7)

for every v ∈ V and w ∈ W, using the Cauchy–Schwarz inequality in the first step. This implies that v ↦ ⟨T (v), w⟩_{W} (21.8)

is a bounded linear functional on V for each w ∈ W, with dual norm less than or equal to ∥T ∥_{op,V W} ∥w∥_{W}. If V is a Hilbert space, then for each w ∈ W there is a unique element T^{∗}(w) of V such that

⟨T (v), w⟩_{W} = ⟨v, T^{∗}(w)⟩_{V} (21.9)

for every v ∈ V, as in Section 17. One can check that T^{∗} is a linear mapping from W into V, because T^{∗}(w) is uniquely determined by (21.9). We also have that ∥T^{∗}(w)∥_{V} ≤ ∥T ∥_{op,V W} ∥w∥_{W} (21.10)

for every w ∈ W, because of the corresponding bound for the dual norm of (21.8). This implies that T^{∗} is bounded as a linear mapping from W into V, with operator norm less than or equal to ∥T ∥_{op,V W}. In fact,

∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W} , (21.11)

because of (21.9) and the characterization of ∥T ∥_{op,V W} as (21.3).

Let us continue to ask that V be a Hilbert space, so that

T ↦ T^{∗} (21.12)

defines a mapping from BL(V, W) into BL(W, V). This mapping is linear in the real case, and conjugate-linear in the complex case. Of course, if W is a Hilbert space, then the adjoint of a bounded linear mapping from W into V is defined as a bounded linear mapping from V into W in the same way. If V and W are Hilbert spaces, and T is a bounded linear mapping from V into W, then T^{∗} is defined as a bounded linear mapping from W into V, and the adjoint (T^{∗})^{∗} of T^{∗} is defined as a bounded linear mapping from V into W. It is easy to see that

(T^{∗})^{∗} = T, (21.13)

directly from the definitions.

Let us suppose that V and W are Hilbert spaces for the rest of the section. Remember that the identity mapping I_{V} on V is bounded as a linear mapping from V into itself. Observe that

I_{V}^{∗} = I_{V} (21.14)

as a bounded linear mapping from V into itself.

Let (Z, ⟨·, ·⟩_{Z}) be another Hilbert space, which is real or complex depending on whether V and W are real or complex. Also let T_{1} be a bounded linear mapping from V into W, and let T_{2} be a bounded linear mapping from W into Z. Thus the adjoint T_{1}^{∗} of T_{1} is defined as a bounded linear mapping from W into V, and the adjoint T_{2}^{∗} of T_{2} is defined as a bounded linear mapping from Z into W. If v ∈ V and z ∈ Z, then we have that

⟨T_{2}(T_{1}(v)), z⟩_{Z} = ⟨T_{1}(v), T_{2}^{∗}(z)⟩_{W} = ⟨v, T_{1}^{∗}(T_{2}^{∗}(z))⟩_{V} , (21.15)

using the definition of T_{2}^{∗} in the first step, and the definition of T_{1}^{∗} in the second step. This implies that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗} (21.16)

as bounded linear mappings from Z into V.

Suppose that T is a one-to-one bounded linear mapping from V onto W whose inverse is bounded as a linear mapping from W onto V. Thus T^{∗} is defined as a bounded linear mapping from W into V, and (T^{−1})^{∗} is defined as a bounded linear mapping from V into W. Observe that

T^{∗} ∘ (T^{−1})^{∗} = (T^{−1} ∘ T)^{∗} = I_{V}^{∗} = I_{V} , (21.17)

using (21.16) in the first step, and (21.14) in the third step. Similarly,

(T^{−1})^{∗} ∘ T^{∗} = (T ∘ T^{−1})^{∗} = I_{W}^{∗} = I_{W} . (21.18)

This implies that T^{∗} is invertible as a bounded linear mapping from W into V, with (T^{∗})^{−1} = (T^{−1})^{∗}. (21.19)

Let T be any bounded linear mapping from V into W again. Remember that T is an isometry if and only if

⟨T (v), T (v′)⟩_{W} = ⟨v, v′⟩_{V} (21.20)

for every v, v′ ∈ V, as in (11.8). This condition is equivalent to asking that

⟨v, T^{∗}(T (v′))⟩_{V} = ⟨v, v′⟩ (21.21) for every v, v′ ∈ V, by definition of the adjoint. It is easy to see that (21.21) holds if and only if T^{∗} ∘ T = I_{V} (21.22)

as linear mappings from V into itself. Thus T is an isometry from V into W if and only if (21.22) holds.

Suppose now that T is a one-to-one bounded linear mapping from V onto W. In this case, (21.22) is the same as saying that

T^{−1} = T^{∗}. (21.23)

Thus T is a unitary mapping from V onto W if and only if (21.23) holds.

Part II

Invertibility and related topics

22 Invertibility of multiplication operators

Let k be a field, let X be a nonempty set, and let a be a k-valued function on X. Thus M_{a}(f) = a f (22.1)

defines a linear mapping from the space c(X, k) of k-valued functions on X into itself, as in Section 19. If a(x) ≠ 0 for every x ∈ X, then M_{a} is invertible on c(X, k), with inverse equal to the multiplication operator M_{1/a} associated to 1/a. If a is any k-valued function on X, then the kernel of M_{a} on c(X, k) consists of the k-valued functions f on X such that

supp f ⊆ {x ∈ X : a(x) = 0}. (22.2)

We also have that

(22.3) supp M_{a}(f) = supp(a f) = (supp a) ∩ (supp f) ⊆ supp a

for every f ∈ c(X, k). More precisely, M_{a} maps c(X, k) onto the space of kvalued functions on X whose support is contained in the support of a. There are analogous statements for M_{a} as a linear mapping from c_{00}(X, k) into itself.

Let | · | be a q-absolute value function on k for some q > 0, and let a be a bounded k-valued function on X. Thus M_{a} defines a bounded linear mapping from ℓ^{r}(X, k) into itself for each r > 0, as in Section 19. As before, the kernel of M_{a} on ℓ^{r}(X, k) consists of the f ∈ ℓ^{r}(X, k) that satisfy (22.2). Using (22.3), we get that M_{a} maps ℓ^{r}(X, k) into the subspace of functions whose support is contained in the support of a. Remember that M_{a} maps c_{0}(X, k) into itself as well under these conditions, for which there are analogous statements.

If a(x) ≠ 0 for each x ∈ X, and if 1/a is bounded on X, then M_{a} has a bounded inverse on ℓ^{r}(X, k) for every r > 0, which is given by M_{1/a}. In this case, M_{1/a} maps c_{0}(X, k) into itself too, as before. In the other direction, suppose that for some r > 0 there is a positive real number c such that

∥M_{a}(f)∥_{r} ≥ c ∥f∥_{r} (22.4)

for every f ∈ ℓ^{r}(X, k). This implies that

|a(y)| ≥ c (22.5)

for every y ∈ X, by taking f to be the function δ_{y} on X that is equal to 1 at y and to 0 on X \ {y}. In particular, this works as well when (22.4) holds for every f ∈ c_{0}(X, k) and r = ∞.

Let us now take k = R or C, with the standard absolute value function, and let (X, A, μ) be a measure space. Also let a be an essentially bounded measurable real or complex-valued function on X, so that the corresponding multiplication operator M_{a} defines a bounded linear mapping from L^{r}(X) into itself for every r > 0, as in Section 19. The kernel of M_{a} on L^{r}(X) consists of the f ∈ L^{r}(X) such that f(x) = 0 for μ-almost every x ∈ X such that a(x) ≠ 0. Similarly, M_{a} maps L^{r}(X) into the subspace of functions that are equal to 0 for μ-almost every x ∈ X such that a(x) = 0. If a(x) ≠ 0 for μ-almost every x ∈ X, and if 1/a is essentially bounded on X, then M_{a} has a bounded inverse on L^{r}(X), which is given by M_{1/a}. In the other direction, suppose that (22.4) holds for some r > 0, c ∈ R_{+}, and every f ∈ L^{r}(X). If r = ∞, then one can check that (22.5) holds for μ-almost every y ∈ X. If every measurable subset of X of positive measure contains a measurable subset of positive finite measure, then the analogous argument works when r < ∞ too.

23 The usual series

Let k be a field, and let V be a vector space over k. Remember that the space L(V) of linear mappings from V into itself is an associative algebra with respect to composition of operators, which may also be expressed using the usual notation for multiplication. Let T be a linear mapping from V into itself, and let T^{j} be the jth power of T with respect to composition for each positive integer j. This is interpreted as being the identity operator I = I_{V} on V when j = 0. It is well known and easy to see that

(I − T) ∑_{j=0}^{l} T^{j} = ( ∑_{j=0}^{l} T^{j}) (I − T) = I − T^{l+1} (23.1)

for each nonnegative integer l.

Suppose now that | · | is a q_{k}-absolute value function on k for some q_{k} > 0, and that N_{V} is a q_{V} -norm on V with respect to | · | on k for some q_{V} > 0. As in Section 9, BL(V) is the algebra of bounded linear mappings on V with respect to N_{V}, and ∥T ∥_{op} denotes the corresponding operator q_{V} -norm. If T is a bounded linear mapping from V into itself, then

∥T^{j}∥_{op} ≤ ∥T ∥_{op}^{j} (23.2) for every j ≥ 1, by (9.13). This also works when j = 0, with the right side interpreted as being equal to 1. Suppose from now on in this section that

∥T ∥_{op} < 1, (23.3)

so that

lim_{j→∞} ∥T^{j}∥_{op} = 0, (23.4)

by (23.2). If q is any positive real number, then ∥T ∥_{op}^{q} < 1, and

∑_{j=0}^{∞} ∥T^{j}∥_{op}^{q} ≤ ∑_{j=0}^{∞} ∥T ∥_{op}^{q j} = (1 − ∥T ∥_{op}^{q})^{−1}, (23.5)

by summing the geometric series in the second step. In particular, the sum on the left converges as an infinite series of nonnegative real numbers under these conditions.

Suppose for the rest of the section that V is complete with respect to the q_{V} -metric associated to N_{V}. This implies that BL(V) is complete with respect to the q_{V} -metric associated to ∥ · ∥_{op}, as in Section 9. Suppose for the moment that q_{V} < ∞, so that ∑_{j=0}^{∞} T^{j} converges q_{V} -absolutely with respect to ∥ · ∥_{op}, as in the previous paragraph. This implies that ∑_{j=0}^{∞} T^{j} converges in BL(V) with respect to ∥ · ∥_{op}, as in Section 12, because of completeness. We also get that

(I − T) ∑_{j=0}^{∞} T^{j} = ( ∑_{j=0}^{∞} T^{j}) (I − T) = I, (23.6)

by taking the limit as l → ∞ in (23.1). It follows that I − T is invertible on V, with

(I − T)^{−1} = ∑_{j=0}^{∞} T^{j}. (23.7)

Note that

∥(I − T)^{−1}∥_{op}^{q_{V}} = ∥∑_{j=0}^{∞} T^{j}∥_{op}^{q_{V}} ≤ ∑_{j=0}^{∞} ∥T^{j}∥_{op}^{q_{V}} ≤ (1 − ∥T ∥_{op}^{q_{V}})^{−1}, (23.8)

using (12.3) in the second step, and (23.5) in the third step.

If q_{V} = ∞, then (23.4) implies that ∑_{j=0}^{∞} T^{j} converges in BL(V) with respect to ∥ · ∥_{op}, as in Section 12 again. It follows that I − T is invertible on V, with inverse given as in (23.7), as before. In this case, we have that

∥(I − T)^{−1}∥_{op} = ∥∑_{j=0}^{∞} T^{j}∥_{op} ≤ sup_{j≥0} ∥T^{j}∥_{op} ≤ 1, (23.9)

using (12.5) in the second step. We also have that

∥I − T ∥_{op} ≤ max(∥I∥_{op}, ∥T ∥_{op}) ≤ 1, (23.10)

because ∥ · ∥_{op} is an ultranorm on BL(V). Thus I − T is an isometry from V onto itself in this situation, as in Section 9.

24 Some consequences

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q_{V} -norm N_{V} with respect to | · | on k for some q_{V} > 0. Thus ∥ · ∥_{op} defines a q_{V} -norm on BL(V), as in Section 9. Suppose that V is complete with respect to the q_{V} -metric associated to N_{V}, so that BL(V) is complete with respect to the q_{V} -metric associated to ∥ · ∥_{op}, as before.

Let T_{1}, T_{2} be bounded linear mappings from V into itself, and suppose that T_{1} has a bounded inverse on V. Observe that

T_{1} − T_{2} = T_{1} ∘ (I − T_{1}^{−1} ∘ T_{2}). (24.1)

If ∥T_{1}^{−1} ∘ T_{2}∥_{op} < 1, (24.2)

then I − T_{2} ∘ T_{1}^{−1} has a bounded inverse on V, as in the previous section. This implies that T_{1} − T_{2} has a bounded inverse on V, with

(T_{1} − T_{2})^{−1} = (I − T_{1}^{−1} ∘ T_{2})^{−1} ∘ T_{1}^{−1}, (24.3)

by (24.1).

Under these conditions, we have that

∥(T_{1} − T_{2})^{−1}∥_{op} ≤ ∥(I − T_{1}^{−1} ∘ T_{2})^{−1}∥_{op} ∥T_{1}^{−1}∥_{op}, (24.4)

as in (9.13). If q_{V} < ∞, then it follows that

∥(T_{1} − T_{2})^{−1}∥_{op} ≤ (1 − ∥T_{1}^{−1} ∘ T_{2}∥_{op}^{q_{V}})^{−1/q_{V}} ∥T_{1}^{−1}∥_{op}, (24.5)

because of (23.8). If q_{V} = ∞, then I − T_{1}^{−1} ∘ T_{2} is an isometry from V onto itself, as in the previous section. This means that (I −T_{1}^{−1}∘T_{2})^{−1} is an isometry from V onto itself too. In particular, this implies that

∥(T_{1} − T_{2})^{−1}∥_{op} = ∥T_{1}^{−1}∥_{op}, (24.6)

using (24.3) again.

Note that ∥T_{1}^{−1} ∘ T_{2}∥_{op} ≤ ∥T_{1}^{−1}∥_{op} ∥T_{2}∥_{op}, (24.7)

as in (9.13). Thus (24.2) holds when the right side of (24.7) is strictly less than 1. In this case, (24.5) implies that

∥(T_{1} − T_{2})^{−1}∥_{op} ≤ (1 − ∥T_{1}^{−1}∥_{op}^{q_{V}} ∥T_{2}∥_{op}^{q_{V}})^{−1/q_{V}} ∥T_{1}^{−1}∥_{op} (24.8)

when q_{V} < ∞.

Suppose now that T_{1} is an isometry from V onto itself, so that T_{1}^{−1} is an isometry on V as well. This implies that

∥T_{1}^{−1} ∘ T_{2}∥_{op} = ∥T_{2}∥_{op}. (24.9)

If ∥T_{2}∥_{op} < 1 and q_{V} = ∞, then

(24.10) T_{1} − T_{2} is an isometry from V onto itself,

because of (24.1) and the fact that I − T_{1}^{−1} ∘ T_{2} is an isometry on V, as before.

Let T be a bounded linear mapping on V, and let a ∈ k be given. Suppose for the moment that a ≠ 0, so that

a I − T = a (I − a^{−1} T). (24.11)

If ∥T ∥_{op} < |a|, (24.12) then ∥a^{−1} T ∥_{op} = |a|^{−1} ∥T ∥_{op} < 1, (24.13) and I − a^{−1} T has a bounded inverse on V, as in the previous section. This implies that a I − T has a bounded inverse on V, with

(a I − T)^{−1} = a^{−1} (I − a^{−1} T)^{−1}, (24.14)

by (24.11). If q_{V} < ∞, then it follows that

(24.15)  ∥(a I − T)^{−1}∥_{op} = |a|^{−1} ∥(I − a^{−1} T)^{−1}∥_{op}
≤ |a|^{−1} (1 − |a|^{−q_{V}} ∥T ∥_{op}^{q_{V}})^{−1/q_{V}} ,

because of (23.8). If q_{V} = ∞, then I − a^{−1} T is an isometry from V onto itself, as before. Of course, this implies that (I − a^{−1} T)^{−1} is an isometry on V too.

Suppose that T has a bounded inverse on V, and let a ∈ k be given again. Observe that (a I − T) = −T ∘ (I − a T^{−1}). (24.16) If ∥a T^{−1}∥_{op} = |a| ∥T^{−1}∥_{op} < 1, (24.17) then I − a T^{−1} has a bounded inverse on V, as in the previous section. This implies that a I − T has a bounded inverse on V, with

(a I − T)^{−1} = −(I − a T^{−1})^{−1} ∘ T^{−1}, (24.18)

because of (24.16). If q_{V} < ∞, then

(24.19) ∥(a I − T)^{−1}∥_{op} ≤ ∥(I − a T^{−1})^{−1}∥_{op} ∥T^{−1}∥_{op}
≤ (1 − |a|^{q_{V}} ∥T^{−1}∥_{op}^{q_{V}})^{−1/q_{V}} ∥T^{−1}∥_{op},

by (9.13) and (23.8). If q_{V} = ∞, then I − a T^{−1} is an isometry on V, as before, so that (I − a T^{−1})^{−1} is an isometry on V as well. This implies that

∥(a I − T)^{−1}∥_{op} = ∥T^{−1}∥_{op}, (24.20)

using (24.18).

Suppose now that T is an isometry from V onto itself, so that T^{−1} is an isometry on V too. In this case, the remarks in the previous two paragraphs imply that a I − T has a bounded inverse on V when |a| ≠ 1. If |a| < 1 and q_{V} = ∞, then a I − T is an isometry on V, because of (24.16) and the fact that I − a T^{−1} is an isometry on V, as before.

25 Eigenvalues and eigenvectors

Let k be a field, let V be a vector space over k, and let T be a linear mapping from V into itself. As usual, a ∈ k is said to be an eigenvalue of T on V if there is a v ∈ V such that v ≠ 0 and

T (v) = a v. (25.1)

In this case, v is said to be an eigenvector of T on V with respect to the eigenvalue a. Of course, v = 0 satisfies (25.1) trivially, and one may include v = 0 as an eigenvector associated to an eigenvalue a.

The collection of v ∈ V that satisfy (25.1) for a given a ∈ k is the same as the kernel of a I − T on V. Thus a ∈ k is an eigenvalue of T on V if and only if the kernel of a I − T on V is not trivial, which means that a I − V is not one-to-one on V. In particular, this implies that a I − V is not invertible as a linear mapping on V. If V has finite dimension, then it is well known that a one-to-one linear mapping from V into itself is surjective, and hence invertible. In this case, a ∈ k is an eigenvalue of T if and only if a I − T is not invertible on V.

Let V^{alg} be the algebraic dual of V, and let T^{alg} be the dual linear mapping from V^{alg} into itself corresponding to T, as in Section 1. If a ∈ k is an eigenvalue of T^{alg} on V^{alg}, then there is a nonzero linear functional λ on V such that

λ ∘ T = T^{alg}(λ) = a λ. (25.2)

Equivalently, this means that

λ ∘ (a I − T) = 0 (25.3)

as a linear functional on V, so that a I − T maps V into the kernel of λ. The condition that λ ≠ 0 as a linear functional on V means that the kernel of λ is a proper linear subspace of V. It follows that a I − T does not map V onto itself when a is an eigenvalue of T^{alg}.

Suppose for the moment that T is a one-to-one linear mapping from V onto itself, so that T is invertible as a linear mapping on V. In particular, this implies that 0 is not an eigenvalue of T on V. Let a ∈ k with a ≠ 0 be given, and observe that (25.1) holds for some v ∈ V if and only if

T^{−1}(v) = (1/a) v. (25.4)

Thus a is an eigenvalue for T on V if and only if 1/a is an eigenvalue for T^{−1} on V. In this case, the corresponding eigenspaces in V are the same.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N_{V} be a q_{V} -norm on V with respect to | · | on k for some q_{V} > 0. This leads to an operator q_{V} -norm ∥ · ∥_{op} on BL(V), as in Section 9. Suppose now that T is a bounded linear mapping from V into itself with respect to N_{V}. If a ∈ k and v ∈ V satisfy (25.1), then we get that

|a| N_{V} (v) = N_{V} (a v) = N_{V} (T (v)) ≤ ∥T ∥_{op} N_{V} (v). (25.5)

This implies that |a| ≤ ∥T ∥_{op} (25.6)

when v ≠ 0.

Similarly, suppose that

c N_{V} (v) ≤ N_{V} (T (v)) (25.7)

for some positive real number c and every v ∈ V. If a ∈ k and v ∈ V satisfy (25.1), then we have that

c N_{V} (v) ≤ N_{V} (T (v)) = N_{V} (a v) = |a| N_{V} (v). (25.8)

It follows that c ≤ |a| (25.9)

when v ≠ 0. In particular, if T has a bounded inverse on V, then (25.7) holds with c = 1/∥T^{−1}∥_{op}, as in Section 9.

Suppose now that T is an isometric linear mapping from V into itself. This implies that T is a bounded linear mapping on V, with ∥T ∥_{op} = 1 when V ≠ {0}. In this case, (25.7) holds with c = 1. If a ∈ k is an eigenvalue of T, then

|a| = 1, (25.10)

by (25.6) and (25.9).

Let V′ be the dual space of bounded linear functionals on V, as in Section 20, with the corresponding dual q_{k}-norm N_{V′}. If T is any bounded linear mapping from V into itself, then the corresponding dual linear mapping T′ is a bounded linear mapping from V′ into itself, as before. Of course, V′ is a linear subspace of V^{alg}, and T′ is the same as the restriction of T^{alg} to V′. If a ∈ k is an eigenvalue of T′, then there is a nonzero bounded linear functional λ on V that satisfies (25.3). This is the same as saying that a I − T maps V into the kernel of λ. Note that the kernel of λ is a closed linear subspace of V, because λ is continuous, and the kernel of λ is a proper linear subspace of V when λ ≠ 0. Thus a I − T maps V into a proper closed linear subspace of V when a is an eigenvalue of T′.

Now let (V, ⟨v, w⟩_{V}) be a real or complex Hilbert space, and let T be a bounded linear mapping from V into itself. Thus the adjoint T^{∗} of T is defined as a bounded linear mapping on V too, as in Section 21. Suppose that a is a real or complex number, as appropriate, which is an eigenvalue of T^{∗}. Let w ∈ V be an eigenvector associated to T^{∗}, so that

T^{∗}(w) = a w. (25.11)

If v ∈ V, then ⟨T (v), w⟩_{V} = ⟨v, T^{∗}(w)⟩_{V} = \overline{a} ⟨v, w⟩_{V} , (25.12)

using (21.9) in the first step. Of course, the complex-conjugate \overline{a} of a is automatically equal to a in the real case. Equivalently,

⟨(\overline{a} I − T)(v), w⟩_{V} = \overline{a}⟨v, w⟩_{V} − ⟨T (v), w⟩_{V} = 0 (25.13)

for every v ∈ V.

26 Eigenfunctions for bilateral shifts

Let k be a field, and let T be the forward shift operator on c(Z, k), as in Section 3. Thus (T (f))(j) = f(j − 1) (26.1)

for every k-valued function f on Z and j ∈ Z, as before. Of course, T is oneto-one on c(Z, k), so that 0 is not an eigenvalue of T. Let a ∈ k with a ≠ 0 be given, and put e_{a}(j) = a^{j} (26.2)

for each j ∈ Z. Observe that

(T (e_{a}))(j) = e_{a}(j − 1) = a^{j−1} = a^{−1} e_{a}(j) (26.3)

for every j ∈ Z, so that T (e_{a}) = a^{−1} e_{a} (26.4)

as elements of c(Z, k). This shows that e_{a} is an eigenvector of T on c(Z, k) with eigenvalue a^{−1}, and it is easy to see that every eigenvector of T on c(Z, k) with eigenvalue a^{−1} is a scalar multiple of e_{a}. Remember that T maps c_{00}(Z, k) into itself. As a mapping on c_{00}(Z, k), T has no eigenvalues.

Note that T (f g) = T (f) T (g) (26.5)

for every f, g ∈ c(Z, k). If f ∈ c(Z, k), a ∈ k, and a ≠ 0, then it follows that

T (e_{a} f) = T (e_{a}) T (f) = a^{−1} e_{a} T (f), (26.6)

by (26.4). This is the same as saying that

T ∘ M_{e_{a}} = a^{−1} M_{e_{a}} ∘ T, (26.7)

where M_{e_{a}} is the multiplication operator on c(Z, k) corresponding to e_{a}, as before. Equivalently, T ∘ M_{e_{a}}^{−1} = a M_{e_{a}}^{−1} ∘ T, (26.8)

which can be obtained directly from (26.7), or using the facts that 1/e_{a} = e_{1/a} and M_{e_{a}}^{−1} = M_{1/e_{a}} = M_{e_{1/a}}. This implies that

M_{e_{a}} ∘ T ∘ M_{e_{a}}^{−1} = a T (26.9)

as linear mappings from c(Z, k) into itself.

Let | · | be a q-absolute value function on k for some q > 0. If a ∈ k and |a| = 1, then |e_{a}(j)| = |a^{j}| = |a|^{j} = 1 (26.10)

for every j ∈ Z_{+}, so that e_{a} ∈ ℓ^{∞}(Z, k). However, if a ∈ k, a ≠ 0, and |a| ≠ 1, then e_{a} is not bounded on Z. It follows that e_{a} is not an element of c_{0}(Z, k) for any a ∈ k \ {0}, and in particular that e_{a} is not an element of ℓ^{r}(Z, k) when 0 < r < ∞. If a ∈ k and |a| = 1, then (26.10) implies that M_{e_{a}} is an isometric linear mapping from ℓ^{r}(Z, k) onto itself for every r > 0, and that M_{e_{a}} maps c_{0}(Z, k) onto itself.

Using (26.9), we get that

M_{e_{a}} ∘ (I − T) ∘ M_{e_{a}}^{−1} = I − M_{e_{a}} ∘ T ∘ M_{e_{a}}^{−1} = I − a T (26.11)

for every a ∈ k with a ≠ 0, as linear mappings from c(Z, k) into itself. We may also consider T and M_{e_{a}} as linear mappings from c_{00}(Z, k) into itself. If |a| = 1, then we can consider T and M_{e_{a}} as linear mappings on ℓ^{r}(Z, k) for any r > 0, or on c_{0}(Z, k), as in the previous paragraph. In these situations, (26.11) permits us to reduce questions about I − a T to the case where a = 1. Thus we shall often focus on I − T in the next sections.

27 Finite support in Z

Let k be a field. If f ∈ c_{00}(Z, k), then

∑_{j=−∞}^{∞} f(j) (27.1)

can be defined as an element of k, by reducing to a finite sum. The mapping from f ∈ c_{00}(Z, k) to the sum (27.1) defines a linear functional on c_{00}(Z, k). We also have that

∑_{j=−∞}^{∞}(T (f))(j) = ∑_{j=−∞}^{∞} f(j − 1) = ∑_{j=−∞}^{∞} f(j), (27.2)

where T is as in (26.1). Thus

∑_{j=−∞}^{∞}(f(j) − (T (f))(j)) = 0 (27.3)

for every f ∈ c_{00}(Z, k), so that I − T maps c_{00}(Z, k) into the kernel of the linear functional on c_{00}(Z, k) defined by (27.1).

If f ∈ c_{00}(Z, k) and j ∈ Z, then put

(R(f))(j) = ∑_{l=0}^{∞} f(j − l), (27.4)

where the sum on the right reduces to a finite sum in k, and hence defines an element of k. This defines R(f) as a k-valued function on Z, and R defines a linear mapping from c_{00}(Z, k) into c(Z, k). Observe that

(R(f))(j) = ∑_{n=−∞}^{∞} f(n) (27.5)

when j ∈ Z is sufficiently large, depending on f ∈ c_{00}(Z, k), because f(n) = 0 when n ∈ Z is sufficiently large. If

∑_{n=−∞}^{∞} f(n) = 0, (27.6)

then it follows that R(f)(j) = 0 when j is sufficiently large. This implies that

R(f) ∈ c_{00}(Z, k) (27.7)

when f ∈ c_{00}(Z, k) satisfies (27.6), because f(n) = 0 when −n is sufficiently large, and hence (R(f))(j) = 0 when −j is sufficiently large.

Let f ∈ c_{00}(Z, k) and j ∈ Z be given, and observe that

(R(T (f)))(j) = ∑_{l=0}^{∞}(T (f))(j − l) = ∑_{l=0}^{∞} f(j − l − 1) (27.8)

and

(T (R(f)))(j) = (R(f))(j − 1) = ∑_{l=0}^{∞} f(j − 1 − l). (27.9)

The sums on the right sides of (27.8) and (27.9) are both equal to

∑_{l=1}^{∞} f(j − l) = (R(f))(j) − f(j). (27.10)

Thus R(T (f)) = T (R(f)) = R(f) − f (27.11)

for every f ∈ c_{00}(Z, k). This shows that

T ∘ R = R ∘ T = R − I (27.12)

as linear mappings from c_{00}(Z, k) into c(Z, k). More precisely, the first T in (27.12) is considered as a mapping from c(Z, k) into itself, the second T is considered as a mapping from c_{00}(Z, k) into itself, and I is the identity mapping on c_{00}(Z, k). It follows that

(I − T) ∘ R = R ∘ (I − T) = I (27.13)

as mappings from c_{00}(Z, k) into c(Z, k), where the first I and T are considered as mappings on c(Z, k), and the other I’s and T are considered as mappings on c_{00}(Z, k). Alternatively,

(R(f))(j) = ∑_{l=0}^{∞}(T^{l}(f))(j) (27.14)

for every f ∈ c_{00}(Z, k) and j ∈ Z, because the lth power T^{l} of T with respect to composition is as in (3.3). Basically, R corresponds to ∑_{l=0}^{∞} T^{l} as a linear mapping from c_{00}(Z, k) into c(Z, k), where this sum is interpreted as in (27.4) and (27.14).

One can check that I −T is injective on c_{00}(Z, k). This is the same as saying that 1 ∈ k is not an eigenvalue of T on c_{00}(Z, k), and indeed T has no eigenvalues on c_{00}(Z, k), as mentioned in the previous section. The injectivity of I − T on c_{00}(Z, k) also follows from the second equality in (27.13). If f ∈ c_{00}(Z, k), then

(I − T)(R(f)) = f, (27.15)

because the left side of (27.13) is equal to the identity as a mapping from c_{00}(Z, k) into c(Z, k). Remember that R(f) has finite support in Z when f satisfies (27.6), as in (27.7). In this case, (27.15) implies that f is in the image of I − T on c_{00}(Z, k). Thus the image of I − T on c_{00}(Z, k) is the same as the kernel of (27.1) as a linear functional on c_{00}(Z, k), because the other inclusion follows from (27.3).

28 Arbitrary functions on Z

Let k be a field, and let f be a k-valued function on Z. If j ∈ Z, then put

(28.1)  (R_{0}(f))(j) = ∑_{l=1}^{j} f(l)  when j ≥ 0

= − ∑_{l=j+1}^{0} f(l) when j ≤ 0,

where both sums are interpreted as being 0 when j = 0. This defines a k-valued function on Z, and R_{0} defines a linear mapping from c(Z, k) into itself. If f has finite support in Z, then (27.4) is the same as

(R(f))(j) = ∑_{l=−∞}^{j} f(l). (28.2)

In this case, we have that

(R_{0}(f))(j) = (R(f))(j) − ∑_{l=−∞}^{0} f(l) (28.3)

for every j ∈ Z, where the sum on the right side of (28.3) reduces to a finite sum in k.

Let T be the forward shift operator on c(Z, k) again, as in (26.1). Let us check that (R_{0}(T (f)))(j) = (R_{0}(f))(j) − f(j) + f(0) (28.4) for every f ∈ c(Z, k) and j ∈ Z. If j = 0, then both sides of (28.4) are equal to 0. If j ≥ 1, then

(R_{0}(T (f))(j) = ∑_{l=1}^{j}(T (f))(l) = ∑_{l=1}^{j} f(l − 1) = ∑_{l=0}^{j−1} f(l)
(28.5)  = (R_{0}(f))(j) − f(j) + f(0).

If j ≤ −1, then

(R_{0}(T (f))(j) = − ∑_{l=j+1}^{0}(T (f))(l)) = − ∑_{l=j+1}^{0} f(l − 1) = − ∑_{l=j}^{−1} f(l)
(28.6)  = (R_{0}(f))(j) − f(j) + f(0).

Similarly, let us check that (T (R_{0}(f)))(j) = (R_{0}(f))(j − 1) = R_{0}(f)(j) − f(j) (28.7) for every f ∈ c(Z, k) and j ∈ Z. Of course, the first step in (28.7) follows from the definition of T. If j ≥ 1, then

(R_{0}(f))(j − 1) = ∑_{l=1}^{j−1} f(l) = (R_{0}(f))(j) − f(j), (28.8)

where the sum in the middle is interpreted as being equal to 0 when j = 1. If j ≤ 0, then

(R_{0}(f))(j − 1) = − ∑_{l=j}^{0} f(l) = (R_{0}(f))(j) − f(j). (28.9)

Equivalently, (28.7) says that T (R_{0}(f)) = R_{0}(f) − f (28.10) for every f ∈ c(Z, k). Thus T ∘ R_{0} = R_{0} − I (28.11) as linear mappings on c(Z, k). This implies that (I − T) ∘ R_{0} = R_{0} − T ∘ R_{0} = I (28.12) as linear mappings on c(Z, k). It follows that I − T maps c(Z, k) onto itself.

Let λ_{0} be the linear functional on c(Z, k) defined by λ_{0}(f) = f(0) (28.13) for every f ∈ c(Z, k). As in Section 19, we let 1_{Z} = 1_{Z,k} be the constant function equal to 1 = 1_{k} ∈ k on Z. Using (28.4), we get that

R_{0}(T (f)) = R_{0}(f) − f + f(0) 1_{Z} (28.14)

for every f ∈ c(Z, k). Hence

R_{0} ∘ T = R_{0} − I + λ_{0} 1_{Z} (28.15)

as linear mappings on c(Z, k). More precisely, f ↦ λ_{0}(f) 1_{Z} defines a linear mapping from c(Z, k) into itself, which sends f ∈ c(Z, k) to the constant function on Z equal to (28.13). It follows that

R_{0} ∘ (I − T) = R_{0} − R_{0} ∘ T = I − λ_{0} 1_{Z} (28.16)

as linear mappings on c(Z, k). Note that the kernel of I − T on c(Z, k) consists of constant functions, which is the same as the kernel of the right side of (28.16).

29 Summable functions on Z

In this section, we take k = R or C, with the standard absolute value function. It will be convenient to let ℓ^{r}(Z) refer to either ℓ^{r}(Z, R) or ℓ^{r}(Z, C) for each r > 0, and similarly for c_{00}(Z), c_{0}(Z), and c(Z).

If f is a summable real or complex-valued function on Z, then

∑_{j=−∞}^{∞} f(j) (29.1)

can be defined as a real or complex number. This may be treated as a sum over Z, as in Section 10, or as a sum of two absolutely convergent infinite series. Of course, the mapping from f ∈ ℓ^{1}(X) to the sum (29.1) defines a bounded linear functional on ℓ^{1}(X) with respect to the ℓ^{1} norm. It is easy to see that (27.2) and (27.3) still hold in this situation, where T is as in (26.1) again. Thus I − T maps ℓ^{1}(X) into the kernel of the linear functional defined by (29.1), as before.

Let f be a summable real or complex-valued function on Z again, and observe that

∑_{l=0}^{∞} f(j − l) (29.2)

converges absolutely for each j ∈ Z. This is the same as the right side of (27.4), so that (R(f))(j) can be defined as before. Equivalently, this sum can be expressed as the right side of (28.2). This implies that

|(R(f))(j)| ≤ ∑_{l=−∞}^{j} |f(l)| ≤ ∥f∥_{1} (29.3)

for every j ∈ Z. Thus R(f) is a bounded real or complex-valued function on Z, as appropriate, with ∥R(f)∥_{∞} ≤ ∥f∥_{1}. (29.4)

Note that

lim_{j→−∞}(R(f))(j) = 0 (29.5)

for every f ∈ ℓ^{1}(Z), by the first inequality in (29.3). If (29.1) is equal to 0, then we have that

(R(f))(j) = − ∑_{l=j+1}^{∞} f(l) (29.6)

for each j ∈ Z. This implies that

|(R(f))(j)| ≤ ∑_{l=j+1}^{∞} |f(l)| (29.7)

for each j ∈ Z. It follows that

lim_{j→∞}(R(f))(j) = 0 (29.8)

under these conditions. Combining (29.5) and (29.8), we get that

R(f) ∈ c_{0}(Z) (29.9)

for every f ∈ ℓ^{1}(Z) such that (29.1) is equal to 0.

If f ∈ ℓ^{1}(Z), then (27.8) and (27.9) hold for every j ∈ Z, for the same reasons as before. This implies that (27.11) holds for every f ∈ ℓ^{1}(Z), so that

T ∘ R = R ∘ T = R − I (29.10)

as linear mappings from ℓ^{1}(Z) into ℓ^{∞}(Z), as in (27.12). More precisely, the first T in (29.10) is considered as a mapping from ℓ^{∞}(Z) into itself, the second T is considered as a mapping from ℓ^{1}(Z) into itself, I is the identity mapping on ℓ^{1}(Z), and R is considered as a mapping from ℓ^{1}(X) into ℓ^{∞}(X). Hence

(I − T) ∘ R = R ∘ (I − T) = I (29.11)

as linear mappings from ℓ^{1}(Z) into ℓ^{∞}(Z), as in (27.13), where the first I and T in (29.11) are considered as mappings on ℓ^{∞}(Z), and the other I’s and T are considered as mappings on ℓ^{1}(Z).

Remember that T has no eigenvalues on ℓ^{1}(Z), as in Section 26. In particular, 1 is not an eigenvalue of T on ℓ^{1}(Z), which means that I − T is injective on ℓ^{1}(Z). This can also be obtained from the second equality in (29.11), as in Section 27. More precisely, if f ∈ ℓ^{1}(Z), then

R(f − T (f)) = f, (29.12) by the second equality in (29.11). This implies that

∥f∥_{∞} = ∥R(f − T (f))∥_{∞} ≤ ∥f − T (f)∥_{1}, (29.13)

using (29.4) in the second step.

Remember too that I − T maps ℓ^{1}(Z) into

{g ∈ ℓ^{1}(Z) : ∑_{j=−∞}^{∞} g(j) = 0}, (29.14)

as mentioned earlier. Note that (29.14) is a closed linear subspace of ℓ^{1}(X) with respect to the metric associated to the ℓ^{1} norm, because it is the kernel of the bounded linear functional defined by (29.1). Of course,

{g ∈ c_{00}(Z) : ∑_{j=−∞}^{∞} g(j) = 0} (29.15)

is a linear subspace of (29.14). One can check that (29.15) is dense in (29.14) with respect to the metric associated to the ℓ^{1} norm. More precisely, c_{00}(Z) is dense in ℓ^{1}(Z), as in Section 8. An additional adjustment is needed to show that (29.15) is dense in (29.14), to get the condition on the sum. We have seen that I − T maps c_{00}(Z) onto (29.15), as in Section 27. It follows that I − T maps ℓ^{1}(Z) onto a dense linear subspace of (29.14).

30 Approximate eigenvalues

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V, W be vector spaces over k with q_{V}, q_{W} -norms N_{V}, N_{W} with respect to | · | on k for some q_{V} , q_{W} > 0, respectively. Also let T be a linear mapping from V into W, and consider the condition that there be a positive real number c such that c N_{V} (v) ≤ N_{W} (T (v)) (30.1)

for every v ∈ V. This implies that the kernel of T is trivial, so that T is injective. If T is a one-to-one linear mapping from V onto W, then this condition is equivalent to the boundedness of the inverse T^{−1} as a linear mapping from W onto V. In this case, (30.1) is the same as saying that

∥T^{−1}∥_{op,W V} ≤ 1/c, (30.2)

where ∥T^{−1}∥_{op,W V} is the operator q_{V} -norm of T^{−1} associated to N_{V} and N_{W}, as in Section 9.

Let us now consider the condition that there is no c > 0 such that (30.1) holds for every v ∈ V. In particular, this implies that T does not have a bounded inverse mapping. This condition is the same as saying that for each ϵ > 0 there is a v_{ϵ} ∈ V such that N_{W} (T (v_{ϵ})) < ϵ N_{V} (v_{ϵ}). (30.3)

Equivalently, this means that there is a sequence {v_{j}}_{j=1}^{∞} of nonzero vectors in V such that

lim_{j→∞} N_{W} (T (v_{j})) N_{V} (v_{j})^{−1} = 0. (30.4)

If the kernel of T is nontrivial, then we can take {v_{j}}_{j=1}^{∞} to be a constant sequence in V.

Let us take V = W for the rest of the section. Let T be a linear mapping from V into itself, and let a ∈ k be given. Consider the condition that

c N_{V} (v) ≤ N_{V} ((a I − T)(v)) = N_{V} (a v − T (v)) (30.5)

for some c > 0 and every v ∈ V. This condition holds when a I − T has a bounded inverse on V, as before. If this condition does not hold, then there is a sequence {v_{j}}_{j=1}^{∞} of nonzero vectors in V such that

lim_{j→∞} N_{V} (a v_{j} − T (v_{j})) N_{V} (v_{j})^{−1} = 0, (30.6)

as in the preceding paragraph. In this case, one may say that a is an approximate eigenvalue of T on V. If a is an eigenvalue of T, then a is an approximate eigenvalue in this sense. If a is an approximate eigenvalue of T, then a I − T does not have a bounded inverse on V.

Let a ∈ k and a sequence {v_{j}}_{j=1}^{∞} of nonzero vectors in V be given. If q_{V} < ∞, then one can check that

|N_{V} (a v_{j})^{q_{V}} − N_{V} (T (v_{j}))^{q_{V}} | ≤ N_{V} (a v_{j} − T (v_{j}))^{q_{V}} (30.7)

for each j, using the q_{V} -norm version of the triangle inequality. Thus (30.6) implies that

lim_{j→∞} |N_{V} (a v_{j})^{q_{V}} − N_{V} (T (v_{j}))^{q_{V}} | N_{V} (v_{j})^{−q_{V}} = 0. (30.8)

Equivalently, this means that

lim_{j→∞}||a|^{q_{V}} N_{V} (v_{j})^{q_{V}} − N_{V} (T (v_{j}))^{q_{V}} | N_{V} (v_{j})^{−q_{V}} = 0. (30.9)

If q_{V} = ∞, then one can use the ultranorm version of the triangle inequality to get that N_{V} (a v_{j}) = N_{V} (T (v_{j})) (30.10)

for every j ≥ 1 such that

N_{V} (a v_{j} − T (v_{j})) < max(N_{V} (a v_{j}), N_{V} (T (v_{j}))). (30.11)

This is the same as saying that

|a| N_{V} (v_{j}) = N_{V} (T (v_{j})) (30.12)

when N_{V} (a v_{j} − T (v_{j})) < max(|a| N_{V} (v_{j}), N_{V} (T (v_{j}))). (30.13)

If a ≠ 0, then (30.6) implies that (30.13) holds for all sufficiently large j. Thus (30.12) holds for all sufficiently large j when a ≠ 0 and (30.9) holds.

Suppose that T is a bounded linear mapping from V into itself, and let ∥T ∥_{op} be the corresponding operator q_{V} -norm of T, as in Section 9. If a ∈ k is an approximate eigenvalue of T on V, then

|a| ≤ ∥T ∥_{op}. (30.14)

To see this, let {v_{j}}_{j=1}^{∞} be a sequence of nonzero vectors in V that satisfies (30.6). Note that N_{V} (T (v_{j})) ≤ ∥T ∥_{op} N_{V} (v_{j}) (30.15)

for each j, by definition of ∥T ∥_{op}. If |a| > ∥T ∥_{op} and q_{V} < ∞, then one can get a contradiction using (30.9) and (30.15). If q_{V} = ∞ and a ≠ 0, then (30.12) holds for all sufficiently large j, as in the preceding paragraph. This implies that (30.14) holds when q_{V} = ∞ and a ≠ 0, and of course (30.14) is trivial when a = 0.

Suppose now that c N_{V} (v) ≤ N_{V} (T (v)) (30.16)

for some c > 0 and every v ∈ V. Let a ∈ k be an approximate eigenvalue of T on V, and let us check that c ≤ |a|. (30.17)

Let {v_{j}}_{j=1}^{∞} be a sequence of nonzero vectors in V that satisfies (30.6) again. It is easy to see directly that a ≠ 0 in this case, which corresponds to the remarks at the beginning of the section. If q_{V} < ∞ and |a| < c, then one can get a contradiction using (30.9) and (30.16). If q_{V} = ∞ and a ≠ 0, then (30.12) holds for all sufficiently large j, as before. Combining this with (30.16), we get (30.17), as desired.

If T is an isometric linear mapping from V into itself, then T is bounded in particular, with ∥T ∥_{op} = 1, unless V = {0}. In this case, (30.16) holds with c = 1 as well. If a ∈ k is an approximate eigenvalue of T on V, then it follows that |a| = 1, by (30.14) and (30.17).

Let T be a one-to-one linear mapping from V onto itself, and suppose that T^{−1} is bounded as a linear mapping on V. This implies that (30.16) holds, with

c = 1/∥T^{−1}∥_{op}, (30.18)

at least if V ≠ {0}, so that ∥T^{−1}∥_{op} > 0. Alternatively, if a ∈ k is an approximate eigenvalue of T, then one can check that a ≠ 0 and 1/a is an approximate eigenvalue of T^{−1}. More precisely, if {v_{j}}_{j=1}^{∞} is a sequence of nonzero vectors in V that satisfies (30.6), then {v_{j}}_{j=1} satisfies the analogous condition for T^{−1} and 1/a. This uses the fact that

(30.19)  N_{V} (a T^{−1}(v_{j}) − v_{j}) = N_{V} (T^{−1}(a v_{j} − T (v_{j})))
≤ ∥T^{−1}∥_{op} N_{V} (a v_{j} − T (v_{j}))

for each j. It follows that

1/|a| = |1/a| ≤ ∥T^{−1}∥_{op}, (30.20)

as in (30.14). This is the same as (30.17), with c as in (30.18).

31 Eigenvalues of multiplication operators

Let k be a field, and let X be a nonempty set. Also let b be a k-valued function on X, and let M_{b}(f) = b f (31.1)

be the corresponding multiplication operator on c(X, k), as in Section 19. If a ∈ k is an eigenvalue of M_{b} on c(X, k), then there is a nonzero k-valued function f on X such that

b(x) f(x) = a f(x) (31.2)

for every x ∈ X. This implies that

b(x) = a (31.3)

for every x ∈ X such that f(x) ≠ 0. In particular, (31.3) holds for some x ∈ X, because f is not identically zero on X. In the other direction, (31.2) holds for every f ∈ c(X, k) whose support is contained in the set of x ∈ X such that (31.3) holds. If (31.3) holds for some x ∈ X, then a is an eigenvalue of M_{b} on c(X, k). In this case, a is an eigenvalue for the restriction of M_{a} to c_{00}(X, k) as well.

Let | · | be a q-absolute value function on k for some q > 0, and suppose that b is a bounded k-valued function on X. Thus M_{b} defines a bounded linear mapping from ℓ^{r}(X, k) into itself for every r > 0, and M_{b} maps c_{0}(X, k) into itself, as in Section 19. As in the previous paragraph, a ∈ k is an eigenvalue of M_{b} on any of these spaces if and only if (31.3) holds for some x ∈ X. In this case, the corresponding eigenfunctions are the functions in the appropriate space that are supported in the set of x ∈ X such that (31.3) holds, as before.

Suppose that a ∈ k is an element of the closure of b(X) in k, with respect to the q-metric associated to | · |. This means that there is a sequence {x_{j}}_{j=1}^{∞} of elements of X such that {b(x_{j})}_{j=1}^{∞} converges to a in k. Let δ_{x_{j}} be the k-valued function on X equal to 1 at x_{j} and 0 everywhere else, as in (2.2). Thus

M_{b}(δ_{x_{j}}) = b(x_{j}) δ_{x_{j}} (31.4)

for each j, as in (19.3), and ∥δ_{x_{j}}∥_{r} = 1 (31.5)

for every j and r > 0. This implies that a is an approximate eigenvalue of M_{b} on ℓ^{r}(X, k) for each r > 0, and also on c_{0}(X, k), using the ℓ^{∞} q-norm on c_{0}(X, k).

Note that M_{a 1_{X}−b} = a I − M_{b} (31.6)

as linear mappings from c(X, k) into itself. If a ∉ b(X), then a 1_{X} −b is nonzero on X, so that 1/(a 1_{X} − b) (31.7)

is defined as a k-valued function on X. The multiplication operator on c(X, k) associated to (31.7) is the inverse of (31.6), as in Section 22. Suppose that a is not in the closure of b(X) in k, so that there is a positive real number c such that |a − b(x)| ≥ c (31.8)

for every x ∈ X. This implies that (31.7) is bounded on X, so that the multiplication operator associated to (31.7) is bounded on ℓ^{r}(X, k) for each r > 0. This means that (31.6) has a bounded inverse on ℓ^{r}(X, k) for each r > 0, and on c_{0}(X, k). It follows that a is not an approximate eigenvalue of M_{b} on ℓ^{r}(X, k) for any r > 0. In particular, a is not an approximate eigenvalue of M_{b} on c_{0}(X, k) with respect to the ℓ^{∞} q-norm.

Let us take k = R or C with the standard absolute value function for the rest of the section, and let (X, A, μ) be a measure space. Let b be an essentially bounded measurable real or complex-valued function on X, so that the corresponding multiplication operator M_{b} defines a bounded linear mapping from L^{r}(X) into itself for each r > 0. If a is a real or complex number, as appropriate, and f ∈ L^{r}(X) for some r > 0, then

M_{b}(f) = a f (31.9)

in L^{r}(X) means that b f = a f almost everywhere on X with respect to μ. If this holds for some f that is not equal to 0 almost everywhere on X, then

μ({x ∈ X : b(x) = a}) > 0. (31.10)

If E ⊆ X is a measurable set such that b = a almost everywhere on E with respect to μ, then (31.9) holds with f = 1_{E}. Here 1_{E} denotes the indicator function on X associated to E, which is equal to 1 on E and to 0 on X \ E. Of course, 1_{E} is not equal to 0 almost everywhere on X with respect to μ when μ(E) > 0. In this case, 1_{E} is a nonzero eigenvector of M_{b} on L^{∞}(X) corresponding to the eigenvalue a. If μ(E) < ∞ too, then 1_{E} is a nonzero eigenvector of M_{b} on L^{r}(X) for every r > 0, corresponding to the eigenvalue a.

A real or complex number a is said to be an element of the essential range of b if μ({x ∈ X : |b(x) − a| < ϵ}) > 0 (31.11)

for every ϵ > 0. Equivalently, this means that for each ϵ > 0 there is a measurable set E_{ϵ} ⊆ X such that μ(E_{ϵ}) > 0 and

μ({x ∈ E_{ϵ} : |b(x) − a| ≥ ϵ}) = 0. (31.12)

Thus |M_{b}(1_{E_{ϵ}}) − a 1_{E_{ϵ}}| ≤ ϵ 1_{E_{ϵ}} (31.13)

almost everywhere on X with respect to μ. This implies that a is an approximate eigenvalue of M_{b} on L^{∞}(X). If we can also choose E_{ϵ} so that μ(E_{ϵ}) < ∞ for each ϵ > 0, then it is easy to see that a is an approximate eigenvalue of M_{b} on L^{r}(X) for every r > 0.

If a real or complex number a, as appropriate, is not in the essential range of b on X, then a 1_{X} − b is nonzero almost everywhere on X with respect to μ, and 1/(a 1_{X} − b) is essentially bounded on X. This implies that a I − M_{b} has a bounded inverse on L^{r}(X) for every r > 0. It follows that a is not an approximate eigenvalue of M_{b} on L^{r}(X) for any r > 0.

32 Eigenvalues of bilateral shifts

Let k be a field, and let | · | be a q-absolute value function on k for some q > 0. Also let T be the forward shift operator on c(Z, k), as in Section 3 and (26.1). The eigenvalues of T on various subspaces of c(Z, k) were discussed in Section 26, and we would like to consider approximate eigenvalues of T on some of these spaces in this section. If a ∈ k and |a| ≠ 1, then a cannot be an approximate eigenvalue of T on ℓ^{r}(Z, k) for any r > 0. This follows from the remarks in Section 30, and the fact that T is an isometry on ℓ^{r}(Z, k) for each r > 0. If |a| = 1, then we have seen that a is an eigenvalue of T on ℓ^{∞}(Z, k). We have also seen that a is not an eigenvalue of T on ℓ^{r}(Z, k) when 0 < r < ∞.

Let a ∈ k with |a| = 1 and a positive real number r be given, and let j_{1}, j_{2} be integers with j_{1} ≤ j_{2}. Let f be the k-valued function defined on Z by

(32.1)  f(j) = a^{−j} when j_{1} ≤ j ≤ j_{2}
= 0  otherwise.

Thus

(32.2) (T (f))(j) = f(j − 1) = a^{1−j} when j_{1} + 1 ≤ j ≤ j_{2} + 1
= 0  otherwise.

It follows that

(32.3)  a f(j) − (T (f))(j) = a^{1−j_{1}}  when j = j_{1}
= −a^{1−j_{2}} when j = j_{2}
= 0  otherwise.

This implies that ∥a f − T (f)∥_{r} = 2^{1/r}. (32.4)

Similarly, ∥f∥_{r} = (j_{2} − j_{1} + 1)^{1/r}. (32.5)

This shows that a is an approximate eigenvalue of T on ℓ^{r}(Z, k), because we can choose j_{1}, j_{2} so that (32.5) is arbitrarily large.

Let us now take k = R or C with the standard absolute value function for the rest of the section. Let a ∈ R or C be given, as appropriate, with |a| = 1, and let a nonnegative integer n be given as well. Consider the real or complex-valued function f_{n}, as appropriate, defined on Z by

(32.6)  f_{n}(j) = a^{−j} (n − |j|) when |j| ≤ n
= 0  when |j| > n.

Note that f_{n}(j) = 0 when |j| = n. Using (32.6), we get that

(32.7) (T (f_{n}))(j) = f_{n}(j − 1) = a^{1−j} (n − |j − 1|) when |j − 1| ≤ n
= 0  when |j − 1| > n,

which is equal to 0 when |j − 1| = n. If −n + 1 ≤ j ≤ n, so that |j| ≤ n and |j − 1| ≤ n, then

(32.8) a f_{n}(j) − (T (f_{n}))(j) = a^{1−j} (n − |j|) − a^{j−1} (n − |j − 1|)
= a^{j−1} (|j − 1| − |j|).

This implies that |a f_{n}(j) − (T (f_{n}))(j)| = 1 (32.9)

when −n + 1 ≤ j ≤ n. Otherwise, if j ≥ n + 1 or j ≤ −n, then f_{n}(j) = f_{n}(j − 1) = 0, which implies that

a f_{n}(j) − (T (f_{n}))(j) = 0. (32.10)

It follows that ∥a f_{n} − T (f_{n})∥_{r} = (2 n)^{1/r} (32.11)

for every r > 0, where the right side of (32.11) is interpreted as being equal to 1 when r = ∞, as usual. Observe that

∥f_{n}∥_{∞} = |f_{n}(0)| = n, (32.12)

and that |f_{n}(j)| = n − |j| ≥ n/2 (32.13)

when |j| ≤ n/2. There are always at least n integers j with |j| ≤ n/2, so that

∥f_{n}∥_{r} ≥ (n/2) n^{1/r} (32.14)

when 0 < r < ∞. Using (32.11) and (32.14), we get that a is an approximate eigenvalue of T on ℓ^{r}(Z, R) and ℓ^{r}(Z, C) when 0 < r < ∞. Similarly, (32.11) and (32.12) imply that a is an approximate eigenvalue of T on c_{0}(Z, R) and c_{0}(Z, C) with respect to the supremum norm.

33 The ultrametric case

Let k be a field, and suppose that | · | is an ultrametric absolute value function on k. Also let f be a k-valued function on Z, and let a ∈ k be given, with |a| = 1. If j_{1}, j_{2} are integers with j_{1} < j_{2}, then

(33.1)  a^{j_{2}} f(j_{2}) − a^{j_{1}} f(j_{1}) = ∑_{j=j_{1}+1}^{j_{2}}(a^{j} f(j) − a^{j−1} f(j − 1))
= ∑_{j=j_{1}+1}^{j_{2}} a^{j−1} (a f(j) − f(j − 1)).

This implies that

|a^{j_{2}} f(j_{2}) − a^{j_{1}} f(j_{1})| ≤ max_{j_{1}+1≤j≤j_{2}} |a f(j) − f(j − 1)|, (33.2)

by the ultrametric version of the triangle inequality. Equivalently,

|a^{j_{2}} f(j_{2}) − a^{j_{1}} f(j_{1})| ≤ max_{j_{1}+1≤j≤j_{2}} |a f(j) − (T (f))(j)|, (33.3)

where T is the forward shift operator on c(Z, k), as in Section 3 and (26.1).

If a f − T (f) is bounded on Z, then |a^{j_{2}} f(j_{2}) − a^{j_{1}} f(j_{1})| ≤ ∥a f − T (f)∥_{∞} (33.4) for every j_{1}, j_{2} ∈ Z, by (33.3). Using the ultrametric version of the triangle inequality again, we get that f is bounded on Z, with

∥f∥_{∞} ≤ max (∥a f − T (f)∥_{∞}, inf_{j∈Z} |f(j)|). (33.5)

In particular, if f vanishes at infinity on Z, then ∥f∥_{∞} ≤ ∥a f − T (f)∥_{∞}. (33.6) This shows that a is not an approximate eigenvalue of T on c_{0}(Z, k) with respect to the supremum ultranorm when |a| = 1. The analogous statement for |a| ≠ 1 follows from the remarks in Section 30, as mentioned in the previous section.

More precisely, if a f − T (f) is bounded on Z, then

sup_{j_{1},j_{2}∈Z} |a^{j_{2}} f(j_{2}) − a^{j_{1}} f(j_{1})| = ∥a f − T (f)∥_{∞}. (33.7)

This uses (33.4) to get that the left side of (33.7) is less than or equal to the right side. The opposite inequality follows directly from the definitions, with j_{2} = j_{1} + 1. Of course, ∥a f − T (f)∥_{∞} ≤ max(∥a f∥_{∞}, ∥T (f)∥_{∞}) = ∥f∥_{∞} (33.8) for every f ∈ ℓ^{∞}(Z, k), by the ultrametric version of the triangle inequality. Combining this with (33.6), we get that

∥a f − T (f)∥_{∞} = ∥f∥_{∞} (33.9)

when f ∈ c_{0}(Z, k).

Let f ∈ ℓ^{∞}(Z, k) be given, and let R_{0}(f) be defined on Z as in (28.1). Using the ultrametric version of the triangle inequality, we get that R_{0}(f) is bounded on Z too, with ∥R_{0}(f)∥_{∞} ≤ ∥f∥_{∞}. (33.10)

In particular, it follows that I − T maps ℓ^{∞}(Z, k) onto itself, by (28.12). We also have that

∥f∥_{∞} = ∥(I − T)(R_{0}(f))∥_{∞} ≤ ∥R_{0}(f)∥_{∞}, (33.11)

using (28.12) in the first step, and (33.8) in the second step, with a = 1. Thus

∥R_{0}(f)∥_{∞} = ∥f∥_{∞}, (33.12)

by (33.10) and (33.12). Remember that

(R_{0}(f))(0) = 0 (33.13)

automatically, by the definition (28.1) of R_{0}(f). In fact, R_{0} maps ℓ^{∞}(Z, k) onto the subspace of bounded k-valued functions on Z that are equal to 0 at 0, because of (28.16).

34 k Complete

Let k be a field with an ultrametric absolute value function | · | again, and suppose that k is complete with respect to the ultrametric associated to | · |. If f ∈ c_{0}(Z, k), then the infinite series

∑_{j=1}^{∞} f(j), ∑_{j=0}^{∞} f(−j) (34.1)

converge in k, as in Section 12. This permits us to define

∑_{j=−∞}^{∞} f(j) (34.2)

as an element of k, by combining the sums in (34.1). We also have that

| ∑_{j=−∞}^{∞} f(j)| ≤ sup_{j∈Z} |f(j)|, (34.3)

because of the analogous statement for the sums in (34.1), as in (12.5). The sum (34.2) can be treated as a sum over Z as in Section 13 as well. Note that the mapping from f ∈ c_{0}(Z, k) to the sum (34.2) is linear in f. This mapping defines a bounded linear functional on c_{0}(Z, k) with respect to the supremum norm, with dual norm equal to 1.

If f ∈ c_{0}(Z, k), then one can check that

∑_{j=−∞}^{∞}(T (f))(j) = ∑_{j=−∞}^{∞} f(j − 1) = ∑_{j=−∞}^{∞} f(j), (34.4)

as in (27.2), where T is the usual forward shift operator. This implies that

∑_{j=−∞}^{∞}(f(j) − (T (f))(j)) = 0, (34.5)

as in (27.3). It follows that I − T maps c_{0}(Z, k) into

{g ∈ c_{0}(Z, k) : ∑_{j=−∞}^{∞} g(j) = 0}. (34.6)

Of course, this is the kernel of the linear functional defined on c_{0}(Z, k) by (34.2), as in the preceding paragraph. Thus (34.6) is a closed linear subspace of c_{0}(Z, k) with respect to the topology determined by the supremum metric, because (34.2) defines a bounded linear functional on c_{0}(Z, k) with respect to the supremum norm, as before.

If f ∈ c_{0}(Z, k) and j ∈ Z, then

∑_{l=0}^{∞} f(j − l) (34.7)

converges in k, as in Section 12 again. Let (R(f))(j) denote the value of this sum for each j ∈ Z, as in (27.4). Equivalently, this sum can be expressed as

∑_{l=−∞}^{j} f(l) (34.8)

for every j ∈ Z, as in (28.2). It follows that

|(R(f))(j)| ≤ sup_{l≤j} |f(l)| ≤ ∥f∥_{∞} (34.9)

for every j ∈ Z, using (12.5) in the first step. Thus R(f) defines a bounded k-valued function on Z, with

∥R(f)∥_{∞} ≤ ∥f∥_{∞}. (34.10)

The first inequality in (34.9) implies that

lim_{j→−∞} |(R(f))(j)| = 0 (34.11)

for every f ∈ c_{0}(Z, k). If (34.2) is equal to 0, then

(R(f))(j) = − ∑_{l=j+1}^{∞} f(l) (34.12)

for each j ∈ Z. This implies that

|(R(f))(j)| ≤ sup_{l≥j+1} |f(l)| (34.13)

for each j ∈ Z, as in (12.5). Hence

lim_{j→∞} |(R(f))(j)| = 0, (34.14)

because f vanishes at infinity on Z. This shows that R(f) ∈ c_{0}(Z, k) (34.15) when f ∈ c_{0}(Z, k) and (34.2) is equal to 0, by combining (34.11) and (34.14).

If f ∈ c_{0}(Z, k), then (27.8) and (27.9) hold for every j ∈ Z, which implies that (27.11) holds in this situation as well. This means that T ∘ R = R ∘ T = R − I (34.16) as linear mappings from c_{0}(Z, k) into ℓ^{∞}(Z, k), as in (27.12). More precisely, the first T in (34.16) is considered as a mapping from ℓ^{∞}(Z, k) into itself, the second T is considered as a mapping from c_{0}(Z, k) into itself, I is the identity mapping on c_{0}(Z, k), and R is considered as a mapping from c_{0}(Z, k) into ℓ^{∞}(Z, k). It follows that (I − T) ∘ R = R ∘ (I − T) = I (34.17) as linear mappings from c_{0}(Z, k) into ℓ^{∞}(Z, k), as in (27.13), where the first I and T are considered as mappings on ℓ^{∞}(Z, k), and the other I’s and T are considered as mappings on c_{0}(Z, k).

The second equality in (34.17) implies that R(f − T (f)) = f (34.18) for every f ∈ c_{0}(Z, k). Thus ∥f∥_{∞} = ∥R(f − T (f))∥_{∞} ≤ ∥f − T (f)∥_{∞} (34.19) for every f ∈ c_{0}(Z, k), using (34.10) in the second step. This gives another way to look at (33.6), with a = 1.

If f ∈ c_{0}(Z, k), then (I − T)(R(f)) = f, (34.20)

because the left side of (34.17) is equal to the identity as a mapping from c_{0}(Z, k) into ℓ^{∞}(Z, k). If we also have that the sum in (34.2) is equal to 0, then R(f) ∈ c_{0}(Z, k), as in (34.15). This shows that f ∈ c_{0}(Z, k) lies in the image of I − T on c_{0}(Z, k) when the sum in (34.2) is equal to 0. We have already seen that I − T maps c_{0}(Z, k) into (34.6), and the previous remarks imply that I − T maps c_{0}(Z, k) onto (34.6).

35 The domain of R

Let k be a field, and let T be the usual forward shift operator on c(Z, k), as in Section 3 and (26.1). If f ∈ c(Z, k), j_{1}, j_{2} ∈ Z, and j_{1} ≤ j_{2}, then

∑_{j=j_{1}}^{j_{2}}(f(j) − (T (f))(j)) = ∑_{j=j_{1}}^{j_{2}} f(j) − ∑_{j=j_{1}}^{j_{2}} f(j − 1)

(35.1)  = ∑_{j=j_{1}}^{j_{2}} f(j) − ∑_{j=j_{1}−1}^{j_{2}−1} f(j) = f(j_{2}) − f(j_{1} − 1).

Let | · | be a q-absolute value function on k for some q > 0. Consider the space c_{R}(Z, k) of k-valued functions f on Z such that

∑_{j=0}^{∞} f(−j) (35.2)

converges in k. This is a linear subspace of c(Z, k), because linear combinations of convergent series converge as well. If f ∈ c_{R}(Z, k), then

lim_{j→−∞} |f(j)| = 0, (35.3)

as in Section 12. If q = ∞, and if k is complete with respect to the ultrametric associated to |·|, then (35.3) implies the convergence of (35.2) in k, as in Section 12 again. Note that T maps c_{R}(Z, k) onto itself.

If f ∈ c_{R}(Z, k), then

∑_{l=0}^{∞} f(j − l) (35.4)

converges in k for every j ∈ Z, and we let (R(f))(j) be the value of this sum, as in (27.4). This defines R as a linear mapping from c_{R}(Z, k) into c(Z, k). Equivalently,

(R(f))(j) = ∑_{l=−∞}^{j} f(l) (35.5)

for every f ∈ c_{R}(Z, k) and j ∈ Z, as in (28.2) and (34.8). It is easy to see that

lim_{j→−∞}(R(f))(j) = 0 (35.6)

in k for every f ∈ c_{R}(Z, k), because of the convergence of the series (35.2).

If f ∈ c_{R}(Z, k), then it is easy to see that (27.8) and (27.9) hold for every j ∈ Z. This implies (27.11), as before, so that

T ∘ R = R ∘ T = R − I (35.7)

as linear mappings from c_{R}(Z, k) into c(Z, k), as in (27.12). As in previous situations, the first T in (35.7) is considered as a mapping from c(Z, k) into itself, the second T is considered as a mapping from c_{R}(Z, k) into itself, I is the identity mapping on c_{R}(Z, k), and R is considered as a mapping from c_{R}(Z, k) into c(Z, k). Thus (I − T) ∘ R = R ∘ (I − T) = I (35.8)

as linear mappings from c_{R}(Z, k) into c(Z, k), as in (27.13), where the first I and T are considered as mappings on c(Z, k), and the other I’s and T are considered as mappings on c_{R}(Z, k).

Let f be a k-valued function on Z that satisfies (35.3). Using (35.1), we get that

∑_{l=j_{1}}^{j}(f(l) − (T (f))(l)) = f(j) − f(j_{1} − 1) → f(j) as j_{1} → −∞ (35.9)

for each j ∈ Z. This implies that

f − T (f) ∈ c_{R}(Z, k), (35.10)

by taking j = 0 in (35.9). We also get that

(R(f − T (f)))(j) = ∑_{l=−∞}^{j}(f(l) − (T (f))(l)) = f(j) (35.11)

for each j ∈ Z, by (35.5). This corresponds to the second equality in (35.8) when f ∈ c_{R}(Z, k).

36 Doubly-infinite sums

Let k be a field with a q-absolute value function | · | for some q > 0 again. If f ∈ c(Z, k) and

∑_{j=1}^{∞} f(j) (36.1)

converges in k, then

lim_{j→∞} |f(j)| = 0, (36.2)

as in Section 12. As before, (36.2) implies that (36.1) converges in k when q = ∞ and k is complete with respect to the ultrametric associated to | · |.

Let c_{S}(Z, k) be the space of k-valued functions f on Z such that (35.2) and (36.1) both converge in k. In this case,

∑_{j=−∞}^{∞} f(j) (36.3)

can be defined as the sum of (35.2) and (36.1). Of course, c_{S}(Z, k) is a linear subspace of the subspace c_{R}(Z, k) of c(Z, k) defined in the previous section. It is easy to see that the usual forward shift operator T maps c_{S}(Z, k) onto itself. If f ∈ c_{S}(Z, k), then one can check that

∑_{j=−∞}^{∞}(T (f))(j) = ∑_{j=−∞}^{∞} f(j), (36.4)

as in (27.2).

Suppose that f ∈ c_{S}(Z, k), so that f ∈ c_{R}(Z, k) in particular, and R(f) can be defined as in the previous section. If (36.3) is equal to 0, then we have that

(R(f))(j) = − ∑_{l=j+1}^{∞} f(l) (36.5)

for each j ∈ Z, because of (35.5). Note that the convergence of the sum on the right side of (36.5) follows from the convergence of (36.1). This implies that

lim_{j→∞}(R(f))(j) = 0 (36.6)

in k under these conditions. Combining (35.6) and (36.6), we get that

R(f) ∈ c_{0}(Z, k). (36.7)

Let f be a k-valued function on Z that satisfies (36.2). Observe that

∑_{l=j}^{j_{2}}(f(j) − (T (f))(j)) = f(j_{2}) − f(j − 1) → −f(j − 1) as j_{2} → ∞ (36.8)

for each j ∈ Z, using (35.1) in the first step. If we take j = 1, then we get that

∑_{l=1}^{∞}(f(l) − (T (f))(l)) = −f(0), (36.9)

and in particular the series on the left converges in k. If f ∈ c_{0}(Z, k), then it follows that f − T (f) ∈ c_{S}(Z, k), (36.10) by combining the previous statement with (35.10). In this case, we also have that

∑_{l=−∞}^{0}(f(l) − (T (f))(l)) = f(0), (36.11)

by (35.9). Combining (36.9) and (36.11), we obtain that

∑_{l=−∞}^{∞}(f(l) − (T (f))(l)) = 0 (36.12)

under these conditions. This could be derived from (36.4) as well when f is an element of c_{S}(Z, k).

37 Bounded partial sums

Let k be a field with a q-absolute value function |·| for some q > 0. If f ∈ c(Z, k), then

∥f∥_{BP S} = ∥f∥_{BP S(Z,k)} = sup {| ∑_{j=j_{1}}^{j_{2}} f(j)| : j_{1}, j_{2} ∈ Z, j_{1} ≤ j_{2}} (37.1)

is defined as a nonnegative extended real number. Let us say that f has bounded partial sums on Z when (37.1) is finite. Observe that ∥f∥_{∞} ≤ ∥f∥_{BP S} (37.2) for every f ∈ c(Z, k), by taking j_{1} = j_{2} in (37.1). Let BP S(Z, k) be the space of f ∈ c(Z, k) with bounded partial sums, so that BP S(Z, k) ⊆ ℓ^{∞}(Z, k), (37.3) by (37.2). One can check that BP S(Z, k) is a linear subspace of ℓ^{∞}(Z, k), and that (37.1) defines a q-norm on BP S(Z, k). In particular, (37.2) implies that ∥f∥_{BP S} > 0 when f(j) ≠ 0 for some j ∈ Z.

Suppose for the moment that q = ∞. In this case, it is easy to see that ∥f∥_{BP S} ≤ ∥f∥_{∞} (37.4) for every f ∈ c(Z, k), by the ultrametric version of the triangle inequality. Of course, (37.2) and (37.4) imply that ∥f∥_{BP S} = ∥f∥_{∞} (37.5) for every f ∈ c(Z, k). Thus BP S(Z, k) = ℓ^{∞}(Z, k) (37.6) in this situation.

Suppose now that q < ∞, and let f ∈ c(Z, k) be given. Observe that

| ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ ( ∑_{j=j_{1}}^{j_{2}} |f(j)|^{q})^{1/q} (37.7)

for every j_{1}, j_{2} ∈ Z with j_{1} ≤ j_{2}. This implies that ∥f∥_{BP S} ≤ ∥f∥_{q}, (37.8) and hence ℓ^{q}(Z, k) ⊆ BP S(Z, k). (37.9) If k = R with the standard absolute value function, and f is a nonnegative real-valued function on Z, then

∥f∥_{BP S} = ∑_{j=−∞}^{∞} f(j) = ∥f∥_{1}. (37.10)

In this case, f has bounded partial sums if and only if f is summable on Z.

Let T be the usual forward shift operator on c(Z, k), as in Section 3 and (26.1). If f ∈ c(Z, k), j_{1}, j_{2} ∈ Z, and j_{1} ≤ j_{2}, then

∑_{j=j_{1}}^{j_{2}}(T (f))(j) = ∑_{j=j_{1}}^{j_{2}} f(j − 1) = ∑_{j=j_{1}−1}^{j_{2}−1} f(j). (37.11)

It follows that ∥T (f)∥_{BP S} = ∥f∥_{BP S}, (37.12) and that T maps BP S(Z, k) onto itself.

38 Normalized partial sums

Let k be a field with a q-absolute value function | · | for some q > 0, and let f ∈ c(Z, k) be given. As before,

∥f∥_{BP S_{+}} = sup {|∑_{j=1}^{j_{2}} f(j)| : j_{2} ∈ Z_{+}} (38.1)

and

∥f∥_{BP S_{−}} = sup {| ∑_{j=j_{1}}^{0} f(j)| : j_{1} ∈ Z, j_{1} ≤ 0} (38.2)

are defined as nonnegative extended real numbers. Put ∥f∥_{BP S_{±}} = max(∥f∥_{BP S_{+}}, ∥f∥_{BP S_{−}}), (38.3) which is also defined as a nonnegative extended real number. Clearly

∥f∥_{BP S_{±}} ≤ ∥f∥_{BP S}, (38.4)

because the partial sums in (38.1) and (38.2) are particular cases of the partial sums in (37.1). Similarly, we would like to estimate ∥f∥_{BP S} in terms of ∥f∥_{BP S_{±}}.

Let j_{1}, j_{2} ∈ Z be given, with j_{1} ≤ j_{2}. If j_{1} ≤ 0 and 1 ≤ j_{2}, then

∑_{j=j_{1}}^{j_{2}} f(j) = ∑_{j=j_{1}}^{0} f(j) + ∑_{j=1}^{j_{2}} f(j). (38.5)

If 1 ≤ j_{1}, then

∑_{j=j_{1}}^{j_{2}} f(j) = ∑_{j=1}^{j_{2}} f(j) − ∑_{j=1}^{j_{1}−1} f(j), (38.6)

where the second sum on the right side is interpreted as being equal to 0 when j_{1} = 1. If j_{2} ≤ 0, then

∑_{j=j_{1}}^{j_{2}} f(j) = ∑_{j=j_{1}}^{0} f(j) − ∑_{j=j_{2}+1}^{0} f(j), (38.7)

where the second sum on the right side is interpreted as being equal to 0 when j_{2} = 0. In each of these three cases, one can check that

| ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ 2^{1/q} ∥f∥_{BP S_{±}}, (38.8)

using the q-absolute value function version of the triangle inequality. Here 2^{1/q} is interpreted as being equal to 1 when q = ∞. This implies that

∥f∥_{BP S} ≤ 2^{1/q} ∥f∥_{BP S_{±}}. (38.9)

Thus f has bounded partial sums on Z if and only if ∥f∥_{BP S_{±}} is finite. It is easy to see that ∥f∥_{BP S_{+}}, ∥f∥_{BP S_{−}} are q-seminorms on BP S(Z, k), and that ∥f∥_{BP S_{±}} is a q-norm on BP S(Z, k). If q = ∞, then

∥f∥_{BP S_{±}} = ∥f∥_{BP S} = ∥f∥_{∞} (38.10)

for every f ∈ c(Z, k), by (37.5), (38.4), and (38.9).

Let f ∈ c(Z, k) be given again, and let R_{0}(f) be the k-valued function defined on Z as in (28.1). By construction,

∥R_{0}(f)∥_{∞} = ∥f∥_{BP S_{±}}. (38.11)

In particular, R_{0}(f) is bounded on Z if and only if f has bounded partial sums on Z. If q = ∞, then (33.12) follows from (38.10) and (38.11).

Suppose that f is an element of the space c_{R}(Z, k) defined in Section 35. This means that the infinite series (35.2) converges in k, which implies that the corresponding sequence of partial sums is bounded. Thus ∥f∥_{BP S_{−}} < ∞. (38.12) If R(f) is defined on Z as in Section 35, then |(R(f))(j)| ≤ ∥f∥_{BP S} (38.13) for every j ∈ Z, by (35.5) and the definition (37.1) of ∥f∥_{BP S}. It follows that ∥R(f)∥_{∞} ≤ ∥f∥_{BP S}. (38.14)

If j_{1}, j_{2} ∈ Z and j_{1} ≤ j_{2}, then

∑_{j=j_{1}}^{j_{2}} f(j) = (R(f))(j_{2}) − (R(f))(j_{1} − 1), (38.15)

by (35.5). This implies that

| ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ 2^{1/q} ∥R(f)∥_{∞}, (38.16)

and hence ∥f∥_{BP S} ≤ 2^{1/q} ∥R(f)∥_{∞}. (38.17) If q = ∞, then we get that ∥R(f)∥_{∞} = ∥f∥_{BP S} = ∥f∥_{∞}, (38.18) using (38.14) and (38.17) in the first step, and (37.5) in the second step.

If f is an element of the space c_{S}(Z, k) defined in Section 36, then f is an element of c_{R}(Z, k), and the infinite series (36.1) converges in k. It follows that the sequence of partial sums corresponding to (36.1) is bounded, so that ∥f∥_{BP S_{+}} < ∞. (38.19) Combining this with (38.12), we get that c_{S}(Z, k) ⊆ BP S(Z, k), (38.20) using also (38.3) and (38.9).

39 Connections with I − T

Let k be a field with a q-absolute value function | · | for some q > 0, and let T be the usual forward shift operator on c(Z, k), as in Section 3 and (26.1). If f ∈ c(Z, k), j_{1}, j_{2} ∈ Z, and j_{1} ≤ j_{2}, then

| ∑_{j=j_{1}}^{j_{2}}(f(j) − (T (f))(j))| ≤ |f(j_{2}) − f(j_{1} − 1)|, (39.1)

by (35.1). This implies that

∥f − T (f)∥_{BP S} = sup_{l_{1},l_{2}∈Z} |f(l_{2}) − f(l_{1})|, (39.2)

where ∥·∥_{BP S} is as in (37.1). It follows that f −T (f) has bounded partial sums on Z if and only if f is bounded on Z. Note that

sup_{l_{1},l_{2}∈Z} |f(l_{2}) − f(l_{1})| ≤ 2^{1/q} ∥f∥_{∞}, (39.3)

and that

∥f∥_{∞} ≤ sup_{l_{1},l_{2}∈Z} |f(l_{2}) − f(l_{1})| (39.4)

when inf_{l∈Z} |f(l)| = 0.

Suppose now that f has bounded partial sums on Z. If R_{0}(f) is defined on Z as in (28.1), then R_{0}(f) is bounded on Z, as in the previous section. We also have that R_{0}(f) − T (R_{0}(f)) = f, (39.5)

by (28.10). This implies that I − T maps ℓ^{∞}(Z, k) onto BP S(Z, k). If q = ∞, then BP S(Z, k) is the same as ℓ^{∞}(Z, k), as in Section 37. In this case, the previous statement is the same as saying that I − T maps ℓ^{∞}(Z, k) onto itself. This was mentioned more directly in Section 33.

Let us take k = R or C with the standard absolute value function, for the rest of the section. Let f be a real or complex-valued function on Z, and let j_{1}, j_{2} ∈ Z be given, with j_{1} ≤ j_{2}. Observe that

\frac{1}{j_{2} − j_{1} + 1} | ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ \frac{1}{j_{2} − j_{1} + 1} ∑_{j=j_{1}}^{j_{2}} |f(j)| ≤ ∥f∥_{∞}. (39.6)

Consider the collection of bounded functions f on Z such that

\frac{1}{j_{2} − j_{1} + 1} ∑_{j=j_{1}}^{j_{2}} f(j) → 0 (39.7)

uniformly as j_{2} − j_{1} → ∞. More precisely, this means that for each ϵ > 0 there should be a nonnegative integer L such that

\frac{1}{j_{2} − j_{1} + 1} | ∑_{j=j_{1}}^{j_{2}} f(j)| < ϵ (39.8)

for every j_{1}, j_{2} ∈ Z such that j_{1} ≤ j_{2} and j_{2} − j_{1} ≥ L. It is easy to see that this is a linear subspace of the space ℓ^{∞}(Z) of bounded real or complex-valued functions on Z, as appropriate. One can also check that this is a closed set in ℓ^{∞}(Z) with respect to the supremum metric. Of course, nonzero constant functions on Z do not have this property.

Note that

\frac{1}{j_{2} − j_{1} + 1} | ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ \frac{1}{j_{2} − j_{1} + 1} ∥f∥_{BP S} (39.9)

for every real or complex-valued function f on Z and j_{1}, j_{2} ∈ Z with j_{1} ≤ j_{2}. This implies that (39.7) holds when f has bounded partial sums on Z. It follows that the space of real or complex-valued functions on Z with bounded partial sums is not dense in ℓ^{∞}(Z) with respect to the supremum metric, by the remarks in the preceding paragraph.

40 Some density conditions

Let k be a field, and let T be the usual forward shift operator on c(Z, k), as in Section 3 and (26.1). Remember that I − T maps c_{00}(Z, k) onto

{f ∈ c_{00}(Z, k) : ∑_{j=−∞}^{∞} f(j) = 0}, (40.1)

as in Section 27. More precisely, I − T maps c_{00}(Z, k) into (40.1), because of (27.3). The fact that I−T maps c_{00}(Z, k) onto (40.1) was obtained from (27.15). In this section, we would like to use this to look at the image of I − T on some other spaces of functions on Z.

Let | · | be a q-absolute value function on k for some q > 0. Remember that c_{00}(Z, k) is dense in ℓ^{r}(Z, k) when 0 < r < ∞, as in Section 8. Similarly, c_{00}(Z, k) is dense in c_{0}(Z, k) with respect to the supremum q-metric.

Suppose first that k = R or C, with the standard absolute value function. If 1 < r ≤ ∞, then there are real or complex-valued functions f on Z, as appropriate, such that f has finite support in Z,

∑_{j=−∞}^{∞} f(j) (40.2)

is any given real or complex number, and ∥f∥_{r} is arbitrarily small. More precisely, if a is any real or complex number, n ∈ Z_{+}, f(j) = a/n for n integers j, and f(j) = 0 otherwise, then (40.2) is equal to a, and

∥f∥_{r} = |a| n^{(1/r)−1} (40.3)

for every r > 0. If r > 1, then (40.3) tends to 0 as n → ∞, as desired. Using this, one can check that (40.1) is dense in c_{00}(Z, k) with respect to the ℓ^{r} norm when r > 1. This implies that (40.1) is dense in ℓ^{r}(Z, k) when 1 < r < ∞, because c_{00}(Z, k) is dense in ℓ^{r}(Z, k), as in the previous paragraph. It follows that I − T maps ℓ^{r}(Z, k) onto a dense linear subspace of itself when 1 < r < ∞, because I − T maps c_{00}(Z, k) onto (40.1). Similarly, (40.1) is dense in c_{0}(Z, k) with respect to the supremum metric, because c_{00}(Z, k) is dense in c_{0}(Z, k) with respect to the supremum metric, as before. Hence I − T maps c_{0}(Z, k) onto a dense linear subspace of itself with respect to the supremum metric.

If 0 < r ≤ 1, then r-summable functions on Z are summable on Z, as in Section 8. If f is a summable real or complex-valued function on Z, then the sum (40.2) can be defined as a real or complex-number, as appropriate. Thus the sum (40.2) can be defined as a real or complex number when f is a real or complex-valued r-summable function on Z and 0 < r ≤ 1. In this case, we have that

| ∑_{j=−∞}^{∞} f(j)| ≤ ∥f∥_{1} ≤ ∥f∥_{r}, (40.4)

using (8.4) in the second step. This implies that (40.2) defines a bounded linear functional on ℓ^{r}(Z, k) when 0 < r ≤ 1, so that

{f ∈ ℓ^{r}(Z, k) : ∑_{j=−∞}^{∞} f(j) = 0} (40.5)

is a closed linear subspace of ℓ^{r}(Z, k) when 0 < r ≤ 1.

Observe that I − T maps ℓ^{r}(Z, k) into (40.5) when 0 < r ≤ 1. This was mentioned in Section 29 when r = 1, which implies the analogous statement for 0 < r ≤ 1. In fact, I − T maps ℓ^{r}(Z, k) onto a dense linear subspace of (40.5) when 0 < r ≤ 1. This was also mentioned in Section 29 when r = 1, and the analogous statement for 0 < r ≤ 1 can be shown in essentially the same way. The main point is that (40.1) is dense in (40.5) with respect to the ℓ^{r} r-norm when 0 < r ≤ 1, as before.

Now let k be a field with an ultrametric absolute value function | · |, and suppose that k is complete with respect to the associated ultrametric. If f is an element of c_{0}(Z, k), then the sum (40.2) can be defined as an element of k, as in Section 34. More precisely, this sum defines a bounded linear functional on c_{0}(Z, k) with respect to the corresponding supremum norm, as before. Thus

{f ∈ c_{0}(Z, k) : ∑_{j=−∞}^{∞} f(j) = 0} (40.6)

is a closed linear subspace of c_{0}(Z, k), with respect to the supremum metric. Remember that I − T maps c_{0}(Z, k) onto (40.6) under these conditions, as in Section 34. One can check directly that (40.1) is dense in (40.6) with respect to the supremum metric.

Let r be a positive real number, and remember that ℓ^{r}(Z, k) is contained in c_{0}(Z, k), as in Section 8. If f ∈ ℓ^{r}(Z, k), then it follows that the sum (40.2) can be defined as an element of k, as before. We also have that

| ∑_{j=−∞}^{∞} f(j)| ≤ ∥f∥_{∞} ≤ ∥f∥_{r}, (40.7)

using (34.3) in the first step, and (8.4) in the second step. Thus the sum (40.2) defines a bounded linear functional on ℓ^{r}(Z, k) as well, which implies that (40.5) is a closed linear subspace of ℓ^{r}(Z, k). Of course, I −T maps ℓ^{r}(Z, k) into (40.5), because I − T maps c_{0}(Z, k) into (40.6), as before.

One can check directly that (40.1) is dense in (40.5) with respect to the ℓ^{r} r-norm in this situation. This is analogous to the corresponding statements for real and complex-valued functions on Z, mentioned earlier. As before, this basically uses the density of c_{00}(Z, k) in ℓ^{r}(Z, k), with an additional adjustment to deal with the condition on the sum. It follows that I − T maps ℓ^{r}(Z, k) onto a dense linear subspace of (40.5) with respect to the ℓ^{r} r-norm, because I − T maps c_{00}(Z, k) onto (40.1).

41 Limits of partial sums

Let k be a field with a q-absolute value function | · | for some q > 0, and let f be a k-valued function on Z. To say that the limit

lim_{{j_{1}→−∞ \atop j_{2}→∞}} ∑_{j=j_{1}}^{j_{2}} f(j) (41.1)

exists and is equal to a ∈ k means that for each ϵ > 0 there is a nonnegative integer L such that

| ∑_{j=j_{1}}^{j_{2}} f(j) − a| < ϵ (41.2)

for every j_{1} ≤ −L and j_{2} ≥ L. It is easy to see that the limit a is unique when it exists, by standard arguments. In this case, (41.1) can be used as another way to define the sum of f(j) over j ∈ Z. If f is in the space c_{S}(Z, k) defined in Section 36, then this limit exists and is equal to the value of the sum defined there. Let c_{S,2}(Z, k) be the space of k-valued functions f on Z such that the limit (41.1) exists. This is a linear subspace of c(Z, k), and the value of the limit (41.1) defines a linear functional on c_{S,2}(Z, k).

Let BP S_{0}(Z, k) be the space of k-valued functions on Z with the following property: for each ϵ > 0 there is a nonnegative integer L such that

| ∑_{j=j_{1}}^{j_{2}} f(j) − ∑_{j=j′_{1}}^{j′_{2}} f(j)| < ϵ (41.3)

for every j_{1}, j_{2}, j′_{1}, j′_{2} ∈ Z such that j_{1}, j′_{1} ≤ −L and j_{2}, j′_{2} ≥ L. This may be considered as the Cauchy condition corresponding to the existence of the limit (41.1). In particular, if the limit (41.1) exists, then this Cauchy condition holds, by standard arguments. If we restrict our attention to j_{1} = j′_{1} in this Cauchy condition, then we get that

| ∑_{j=j_{2}+1}^{j′_{2}} f(j)| < ϵ (41.4)

when j′_{2} > j_{2} ≥ L. Similarly, if we restrict our attention to j_{2} = j′_{2}, then we get that

| ∑_{j=j′_{1}}^{j_{1}−1} f(j)| < ϵ (41.5)

when j′_{1} < j_{1} ≤ −L. Conversely, if these two conditions are satisfied, then one can check that the earlier Cauchy condition holds too. More precisely, one can use (41.4) and (41.5) to get a condition like (41.3), with an extra factor of 2^{1/q} on the right side.

The condition (41.4) is equivalent to saying that the sequence of partial sums corresponding to the infinite series ∑_{j=1}^{∞} f(j) is a Cauchy sequence in k. Similarly, the condition (41.5) is equivalent to saying that the sequence of partial sums corresponding to ∑_{j=0}^{∞} f(−j) is a Cauchy sequence. If k is complete with respect to the q-metric associated to | · |, then these Cauchy conditions imply that the two series converge in k. This means that f is in the space c_{S}(Z, k) defined in Section 36. In particular, this implies that the limit (41.1) exists in k, as before.

If f ∈ BP S_{0}(Z, k), then one can check that f has bounded partial sums on Z, as in Section 37. More precisely, BP S_{0}(Z, k) is a closed linear subspace of BP S(Z, k), with respect to the BPS q-norm. In fact, BP S_{0}(Z, k) is the closure of c_{00}(Z, k) in BP S(Z, k). In particular, BP S_{0}(Z, k) is contained in c_{0}(Z, k). If q = ∞, then BP S_{0}(Z, k) is the same as c_{0}(Z, k).

If f ∈ c_{S,2}(Z, k), then

| lim_{{j_{1}→−∞ \atop j_{2}→∞}} ∑_{j=j_{1}}^{j_{2}} f(j)| ≤ ∥f∥_{BP S}, (41.6)

where ∥f∥_{BP S} is the BPS q-norm of f, as in (37.1). Thus the mapping from f to the value of the limit (41.1) defines a bounded linear functional on c_{S,2}(Z, k) with respect to the BPS q-norm. The kernel of this linear functional is the collection of k-valued functions f on Z such that

lim_{{j_{1}→−∞ \atop j_{2}→∞}} ∑_{j=j_{1}}^{j_{2}} f(j) = 0. (41.7)

The boundedness of this linear functional implies that its kernel is relatively closed in c_{S,2}(Z, k), with respect to the topology determined by the BPS qnorm. In fact, one can check that the collection of k-valued functions f on Z for which (41.7) holds is a closed linear subspace of BP S(Z, k), with respect to the topology determined by the BPS q-norm.

42 The inverse of I − a T

Let k be a field with a q-absolute value function | · | for some q > 0, and suppose that k is complete with respect to the associated q-metric. Also let T be the usual forward shift operator on c(Z, k), as in Section 3 and (26.1). Of course, T maps ℓ^{r}(Z, k) isometrically onto itself for each r > 0. If a ∈ k and |a| < 1, then I − a T has a bounded inverse on ℓ^{r}(Z, k) for every r > 0, as in Section 23. The inverse is given by

(I − a T)^{−1} = ∑_{l=0}^{∞} a^{l} T^{l}, (42.1)

where the series converges in BL(ℓ^{r}(Z, k)), as before.

In particular, if f ∈ ℓ^{r}(Z, k) for some r > 0, then

((I − a T)^{−1})(f) = ∑_{l=0}^{∞} a^{l} T^{l}(f), (42.2)

where the series converges in ℓ^{r}(Z, k). It follows that

(((I − a T)^{−1})(f))(j) = ∑_{l=0}^{∞} a^{l} (T^{l}(f))(j) (42.3)

for each j ∈ Z, where the series converges in k. Equivalently,

(((I − a T)^{−1})(f))(j) = ∑_{l=0}^{∞} a^{l} f(j − l) (42.4)

for each j ∈ Z, using (3.3) on the right side. The convergence of the series on the right side of (42.4) in k for each j ∈ Z can be obtained directly from the remarks in Section 12 when |a| < 1 and f is bounded on Z. Similarly, let us look more directly at some of the properties of the linear mapping defined by this series.

Suppose for the moment that r ∈ R_{+}, r ≤ q, and f ∈ ℓ^{r}(Z, k). Remember that | · | can also be considered as an r-absolute value function on k, because r ≤ q. Thus

|∑_{l=0}^{∞} a^{l} f(j − l)|^{r} ≤ ∑_{l=0}^{∞} |a|^{l r} |f(j − l)|^{r} (42.5)

for every j ∈ Z, as in (12.3). This implies that

∑_{j=−∞}^{∞}|∑_{l=0}^{∞} a^{l} f(j − l)|^{r} ≤ ∑_{j=−∞}^{∞} ∑_{l=0}^{∞} |a|^{l r} |f(j − l)|^{l}. (42.6)

Interchanging the order of summation, we get that this double sum is equal to

∑_{l=0}^{∞} ∑_{j=−∞}^{∞} |a|^{l r} |f(j − l)|^{r} = ∑_{l=0}^{∞} ∑_{j=−∞}^{∞} |a|^{l r}|f(j)|^{r} = (1 − |a|^{r})^{−1} ∥f∥_{r}^{r}. (42.7)

This corresponds to (23.8), with V = ℓ^{r}(Z, k) and q_{V} = r. Note that the previous inequalities are equalities when f(j) = 0 for all but one j ∈ Z. This implies that the operator r-norm of (42.1) on ℓ^{r}(Z, k) is equal to

(1 − |a|^{r})^{−1/r} (42.8)

for every a ∈ k with |a| < 1 when r ≤ q.

Suppose now that k = R or C, with the standard absolute value function, and that 1 ≤ r ≤ ∞. In this case, the operator norm of (42.1) on ℓ^{r}(Z, k) is equal to (1 − |a|)^{−1} (42.9)

when |a| < 1. The fact that the operator norm is less than or equal to (42.9) corresponds to (23.8) again, with V = ℓ^{r}(Z, k) and q_{V} = 1. Of course, if a = 0, then (42.1) is the identity operator, which has operator norm 1. Otherwise, suppose that a ≠ 0, and let b be the real or complex number, as appropriate, such that a b = |a|. Thus |b| = 1, so that b is an approximate eigenvalue of T on ℓ^{r}(Z, k), as in Section 32. This implies that a b is an approximate eigenvalue of a T on ℓ^{r}(Z, k), so that 1 − a b is an approximate eigenvalue of I − a T on ℓ^{r}(Z, k). It follows that

(1 − |a|)^{−1} = |1 − a b|^{−1} (42.10)

is less than or equal to the operator norm of (42.1) on ℓ^{r}(Z, k), as in (30.20). Thus the operator norm of (42.1) on ℓ^{r}(Z, k) is equal to (42.9), as desired. Note that b is an eigenvalue of T on ℓ^{∞}(Z, k), as in Section 26, so that 1 − a b is an eigenvalue of I − a T on ℓ^{∞}(Z, k). This does not work for the restriction of T to c_{0}(Z, k), but b is an approximate eigenvalue for the restriction of T to c_{0}(Z, k) with respect to the supremum norm, as in Section 32 again. Hence the operator norm of (42.1) on c_{0}(Z, k) with respect to the supremum norm is equal to (42.9) when |a| < 1, by the same type of argument as before.

Part III

Unilateral shift operators

43 Forward and backward shifts

Let k be a field, and let Z_{0+} = Z_{+} ∪ {0} be the set of nonnegative integers. If f is a k-valued function on Z_{0+}, then let A(f) be the k-valued function defined on Z_{0+} by

(43.1)  (A(f))(j) = f(j − 1) when j ≥ 1
= 0  when j = 0.

This defines a linear mapping A from c(Z_{0+}, k) into itself, which is the forward shift operator on c(Z_{0+}, k). Similarly, let B(f) be the k-valued function defined on Z_{0+} by (B(f))(j) = f(j + 1) (43.2)

for every j ≥ 0. This defines a linear mapping B from c(Z_{0+}, k) into itself, which is the backward shift operator on c(Z_{0+}, k).

More precisely, A is a one-to-one linear mapping from c(Z_{0+}, k) into itself, and B maps c(Z_{0+}, k) onto itself. If f is any k-valued function on Z_{0+}, then

(B(A(f)))(j) = f(j) (43.3)

for every j ∈ Z_{0+}. Thus B ∘ A = I (43.4)

as linear mappings from c(Z_{0+}, k) into itself. Similarly,

(43.5)  (A(B(f)))(j) = f(j) when j ≥ 1
= 0  when j = 0

for every f ∈ c(Z_{0+}, k). Note that A maps c(Z_{0+}, k) onto the space of k-valued functions f on Z_{0+} such that f(0) = 0, and the kernel of B consists of the k-valued functions f on Z_{0+} whose support is contained in {0}.

Let l be a positive integer, and let A^{l}, B^{l} be the lth powers of A, B as linear mappings from c(Z_{0+}, k) into itself with respect to composition, as usual. If f ∈ c(Z_{0+}, k), then

(43.6)  (A^{l}(f))(j) = f(j − l) when j ≥ l
= 0  when 0 ≤ j ≤ l − 1.

Similarly, (B^{l}(f))(j) = f(j + l) (43.7)

for every j ≥ 0. In particular,

B^{l} ∘ A^{l} = I (43.8)

as linear mappings from c(Z_{0+}, k) into itself, which can also be obtained from (43.4). If f ∈ c(Z_{0+}, k) again, then

(43.9)  (A^{l}(B^{l}(f)))(j) = f(j) when j ≥ l
= 0  when 0 ≤ j ≤ l − 1.

Let n ∈ Z_{0+} be given, and let δ_{n}(j) = δ_{Z_{0+},n}(j) be the k-valued function defined on Z_{0+} as in (2.2), which is equal to 1 when j = n and to 0 otherwise. Also let l ∈ Z_{+} be given, and observe that

(43.10) (A^{l}(δ_{n}))(j) = δ_{n}(j − l) = δ_{n+l}(j) when j ≥ l
= 0 = δ_{n+l}(j)  when 0 ≤ j ≤ l − 1.

This implies that A^{l}(δ_{n}) = δ_{n+l}. (43.11)

If j ∈ Z_{0+}, then

(43.12)  (B^{l}(δ_{n}))(j) = δ_{n}(j + l) = δ_{n−l}(j) when l ≤ n
= 0  when l > n.

It follows that

(43.13)  B^{l}(δ_{n}) = δ_{n−l} when l ≤ n
= 0  when l > n.

Let T be the usual forward shift operator on c(Z, k), as in Section 3. Also let f be a k-valued function on Z such that

f(j) = 0 (43.14)

when j < 0, and let f_{0} be the restriction of f to Z_{0+}. If l ∈ Z_{+}, then

(T^{l}(f))(j) = (A^{l}(f_{0}))(j) (43.15)

for every j ≥ 0. Thus A^{l}(f_{0}) is the same as the restriction of T^{l}(f) to Z_{0+}. Note that (T^{l}(f))(j) = f(j − l) = 0 when j < 0, l ≥ 1, and f satisfies (43.14). Similarly, (T^{−l}(f))(j) = f(j + l) = (B^{l}(f_{0}))(j) (43.16)

for every j ≥ 0 and l ≥ 1, so that B^{l}(f_{0}) is the same as the restriction of T^{−l}(f) to Z_{0+}. More precisely, (43.16) holds for every f ∈ c(Z, k), j ≥ 0, and l ≥ 1, without the additional condition (43.14).

44 Polynomials and power series

Let k be a field, and let X be an indeterminate. As in [4, 7], we use upper-case letters like X for indeterminates, and lower-case letters like x for elements of k. If f ∈ c(Z_{0+}, k), then

F (X) = ∑_{j=0}^{∞} f(j) X^{j} (44.1)

is a formal power series in X with coefficients in k. The space of formal power series in X with coefficients in k is typically denoted k[[X]]. Of course, a formal power series is characterized by its coefficients, so that the mapping from a k-valued function f on Z_{0+} to F (X) is a one-to-one correspondence between c(Z_{0+}, k) and k[[X]]. Thus one may use c(Z_{0+}, k) as a precise definition of k[[X]], and use (44.1) as notation for elements of k[[X]]. Note that k[[X]] is a vector space over k with respect to termwise addition and scalar multiplication of formal power series, which correspond exactly to pointwise addition and scalar multiplication of k-valued functions on Z_{0+}.

There is a natural way to multiply formal power series, where

X^{j} X^{l} = X^{j+l} (44.2) for all nonnegative integers j, l. It is well known and easy to see that k[[X]] is a commutative algebra over k with respect to multiplication. Let f ∈ c(Z_{0+}, k) be given, and let A(f) be as defined in (43.1). Thus f and g = A(f) determine formal power series F (X) and G(X), as in (44.1). Observe that

(44.3)  G(X) = ∑_{j=0}^{∞}(A(f))(j) X^{j} = ∑_{j=1}^{∞} f(j − 1) X^{j}

= ∑_{j=0}^{∞} f(j) X^{j+1} = F (X) X.

A formal polynomial in X with coefficients in k may be considered as a formal power series in which all but finitely many coefficients are equal to 0. If f ∈ c_{00}(Z_{0+}, k), then the corresponding formal power series (44.1) is a formal polynomial. The space of formal polynomials in X with coefficients in k is typically denoted k[X], and is a subalgebra of k[[X]]. As before, one can use c_{00}(Z_{0+}, k) as a precise definition of k[X]. Note that the shift operators A, B map c_{00}(Z_{0+}, k) into itself.

If f ∈ c(Z, k), then

F (X) = ∑_{j=−∞}^{∞} f(j) X^{j} (44.4)

may be considered as a formal Laurent series in X with coefficients in k. As usual, one can use c(Z, k) as a precise definition of the space of formal Laurent series in X with coefficients in k. Pointwise addition and scalar multiplication of k-valued functions on Z corresponds to termwise addition and scalar multiplication of formal Laurent series, by construction. Although the product of two formal Laurent series is not always defined, it can be defined in some situations. In particular, it is easy to multiply a formal Laurent series F (X) with a monomial X^{l} for any l ∈ Z. Let f ∈ c(Z, k) be given, and let g = T (f) be as in (3.1). If F (X) and G(X) are the corresponding Laurent series, as in (44.4), then

(44.5) G(X) = ∑_{j=−∞}^{∞}(T (f))(j) X^{j} = ∑_{j=−∞}^{∞} f(j − 1) X^{j}

= ∑_{j=−∞}^{∞} f(j) X^{j+1} = F (X) X.

Of course, one can identify formal power series in X with formal Laurent series in X such that the coefficient of X^{j} is equal to 0 when j < 0.

45 Extension and restriction mappings

Let k be a field, let X be a nonempty set, and let Y be a nonempty subset of X. If f is a k-valued function on X, then let R_{Y} (f) be the restriction of f to Y. Thus R_{Y} defines a linear mapping from c(X, k) onto c(Y, k). Note that R_{Y} maps c_{00}(X, k) onto c_{00}(Y, k).

If f_{0} is a k-valued function on Y, then let E_{Y} (f_{0}) be the k-valued function on X defined by

(45.1)  (E_{Y} (f_{0}))(x) = f_{0}(x) when x ∈ Y
= 0  when x ∈ X \ Y.

This defines a one-to-one linear mapping from c(Y, k) into c(X, k). More precisely, E_{Y} maps c(Y, k) onto the linear subspace

c_{Y} (X, k) = {f ∈ c(X, k) : supp f ⊆ Y } (45.2)

of c(X, k), and E_{Y} (c_{00}(Y, k)) = c_{Y} (X, k) ∩ c_{00}(X, k). (45.3)

Of course, R_{Y} (E_{Y} (f_{0})) = f_{0} (45.4)

for every f_{0} ∈ c(Y, k), so that R_{Y} ∘ E_{Y} is the identity mapping on c(Y, k). If f ∈ c(X, k), then let P_{Y} (f) be the k-valued function defined on X by

(45.5)  (P_{Y} (f))(x) = f(x) when x ∈ Y
= 0  when x ∈ X \ Y.

This defines a linear mapping from c(X, k) onto c_{Y} (X, k), which maps c_{00}(X, k) onto (45.3). Observe that P_{Y} ∘ P_{Y} = P_{Y} , (45.6)

so that P_{Y} defines a projection on c(X, k). We also have that

E_{Y} (R_{Y} (f)) = P_{Y} (f) (45.7)

for every f ∈ c(X, k), so that E_{Y} ∘R_{Y} = P_{Y} as linear mappings on c(X, k). Note that P_{Y} is the same as the multiplication operator on c(X, k) corresponding to the k-valued function on X that is equal to 1 on Y and to 0 on X \ Y, as in Section 19.

Let us now take X = Z and Y = Z_{0+}, so that the restriction operator R_{Z_{0+}}, the extension operator E_{Z_{0+}}, and the projection P_{Z_{0+}} can be defined as before. Also let A, B be the forward and backward shift operators on c(Z_{0+}, k), as in Section 43, and let T be the forward shift operator on c(Z, k), as in Section 3. The condition (43.14) means that f ∈ c_{Z_{0+}}(Z, k), using the notation in (45.2). Thus (43.15) says that

R_{Z_{0+}}(T^{l}(f)) = A^{l}(R_{Z_{0+}}(f)) (45.8)

for every f ∈ c_{Z_{0+}}(Z, k) and l ≥ 1. Similarly,

R_{Z_{0+}}(T^{−l}(f)) = B^{l}(R_{Z_{0+}}(f)) (45.9) for every f ∈ c(Z, k) and l ≥ 1, as in (43.16).

Observe that T (c_{Z_{0+}}(Z, k)) ⊆ c_{Z_{0+}}(Z, k). (45.10)

If f_{0} ∈ c(Z_{0+}, k), then f = E_{Z_{0+}}(f_{0}) ∈ c_{Z_{0+}}(Z, k), and

T^{l}(E_{Z_{0+}}(f_{0})) = E_{Z_{0+}}(A^{l}(f_{0})) (45.11)

for every l ≥ 1. This is a more precise version of (43.15) and (45.8). If we identify c(Z_{0+}, k) with c_{Z_{0+}}(Z, k) using E_{Z_{0+}}, then A^{l} corresponds to the restriction of T^{l} to c_{Z_{0+}}(Z, k) for each l ≥ 1. We also have that

P_{Z_{0+}}(T^{−l}(E_{Z_{0+}}(f_{0}))) = E_{Z_{0+}}(B^{l}(f_{0})) (45.12)

for every f_{0} ∈ c(Z_{0+}, k) and l ≥ 1, as in (43.16) and (45.9).

46 Dual mappings

Let k be a field. If f ∈ c_{00}(Z_{0+}, k) and g ∈ c(Z_{0+}, k), then

λ_{g}(f) = ∑_{j=0}^{∞} f(j) g(j) (46.1)

reduces to a finite sum in k. This defines a linear functional on c_{00}(Z_{0+}, k) for each g ∈ c(Z_{0+}, k), and every linear functional on c_{00}(Z_{0+}, k) is of this form, as in Section 2. Thus g ↦ λ_{g} (46.2)

defines an isomorphism between c(Z_{0+}, k) and the algebraic dual c_{00}(Z_{0+}, k)^{alg} of c_{00}(Z_{0+}, k) as vector spaces over k, as before.

Let A, B be the forward and backward shift operators on c(Z_{0+}, k), as in Section 43. Remember that A, B map c_{00}(Z_{0+}, k) into itself. If f ∈ c_{00}(Z_{0+}, k) and g ∈ c(Z_{0+}, k), then

λ_{g}(A(f)) = ∑_{j=1}^{∞} f(j − 1) g(j), (46.3)

by the definition (43.1) of A(f). It follows that

λ_{g}(A(f)) = ∑_{j=0}^{∞} f(j) g(j + 1) = ∑_{j=0}^{∞} f(j) (B(g))(j) = λ_{B(g)}(f) (46.4)

using the definition (43.2) of B in the second step. This shows that the algebraic dual A^{alg} of A on c_{00}(Z_{0+}, k) corresponds to B on c(Z_{0+}, k), with respect to the isomorphism between c_{00}(Z_{0+}, k)^{alg} and c(Z_{0+}, k) mentioned in the previous paragraph. Similarly,

λ_{g}(B(f)) = ∑_{j=0}^{∞} f(j + 1) g(j) (46.5)

for every f ∈ c_{00}(Z_{0+}, k) and g ∈ c(Z_{0+}, k), by the definition (43.2) of B(f). Hence

λ_{g}(B(f)) = ∑_{j=1}^{∞} f(j) g(j − 1) = ∑_{j=0}^{∞} f(j) (A(g))(j) = λ_{A(g)}(f), (46.6)

using the definition (43.1) of A in the second step. Thus the algebraic dual B^{alg} of B on c_{00}(Z_{0+}, k) corresponds to A on c(Z_{0+}, k), in the same way as before.

Let X be a nonempty set, and let Y be a nonempty subset of X. The corresponding projection operator P_{Y} defined in (45.5) maps c(X, k) into itself, and c_{00}(X, k) into itself. If f ∈ c_{00}(X, k) and g ∈ c(X, k), then

∑_{x∈X}(P_{X}(f))(x) g(x) = ∑_{x∈Y} f(x) g(x) = ∑_{x∈X} f(x) (P_{Y} (g))(x). (46.7)

This implies that the algebraic dual (P_{Y})^{alg} of P_{Y} on c_{00}(X, k) corresponds to P_{Y} on c(X, k), with respect to the usual identification between the algebraic dual c_{00}(X, k)^{alg} of c_{00}(X, k) and c(X, k), as in Section 2.

The restriction operator R_{Y} defined in the previous section maps c(X, k) into c(Y, k), and c_{00}(X, k) into c_{00}(Y, k). The extension operator E_{Y} defined in (45.1) maps c(Y, k) into c(X, k), and c_{00}(Y, k) into c_{00}(X, k). If f ∈ c_{00}(X, k) and g_{0} ∈ c(Y, k), then

∑_{x∈Y}(R_{Y} (f))(x) g_{0}(x) = ∑_{x∈Y} f(x) g_{0}(x) = ∑_{x∈X} f(x) (E_{Y} (g_{0}))(x). (46.8)

This means that the algebraic dual (R_{Y})^{alg} or R_{Y} as a linear mapping from c_{00}(X, k) into c_{00}(Y, k) corresponds to E_{Y} as a linear mapping from c(Y, k) into c(X, k). This uses the identification of the algebraic dual c_{00}(X, k)^{alg} of c_{00}(X, k) with c(X, k) discussed in Section 2, and the analogous identification of the algebraic dual c_{00}(Y, k)^{alg} or c_{00}(Y, k) with c(Y, k). If f_{0} ∈ c_{00}(Y, k) and g ∈ c(X, k), then

∑_{x∈X}(E_{Y} (f_{0}))(x) g(x) = ∑_{x∈Y} f_{0}(x) g(x) = ∑_{x∈Y} f_{0}(x) (R_{Y} (g))(x). (46.9)

This implies that the algebraic dual (E_{Y})^{alg} of E_{Y} as a linear mapping from c_{00}(Y, k) into c_{00}(X, k) corresponds to R_{Y} as a linear mapping from c(X, k) into c(Y, k), using the same identifications of the algebraic dual spaces as before.

47 Boundedness on ℓ^{r} spaces

Let k be a field with a q-absolute value function | · | for some q > 0, and let r > 0 be given. If f ∈ ℓ^{r}(Z_{0+}, k), then it is easy to see that

A(f), B(f) ∈ ℓ^{r}(Z_{0+}, k), (47.1) where A(f) is as in (43.1), and B(f) is as in (43.2). More precisely,

∥A(f)∥_{r} = ∥f∥_{r} (47.2)

for every f ∈ ℓ^{r}(Z_{0+}, k), so that A defines an isometric linear mapping from ℓ^{r}(Z_{0+}, k) into itself. We also have that

∥B(f)∥_{r} ≤ ∥f∥_{r} (47.3)

for every f ∈ ℓ^{r}(Z_{0+}, k), so that B defines a bounded linear mapping from ℓ^{r}(Z_{0+}, k) into itself, with operator q or r-norm less than or equal to 1. If l ∈ Z_{+} and f ∈ ℓ^{r}(Z_{0+}, k) satisfies f(j) = 0 when 0 ≤ j < l, then one can check that ∥B^{l}(f)∥_{r} = ∥f∥_{r}, (47.4)

using (43.7). In particular, this implies that the operator q or r-norm of B^{l} on ℓ^{r}(Z_{0+}, k) is equal to 1 for every l ≥ 1. Note that B maps ℓ^{r}(Z_{0+}, k) onto itself.

Similarly, A maps c_{0}(Z_{0+}, k) into itself, and B maps c_{0}(Z_{0+}, k) onto itself. If l ∈ Z_{+}, then the restriction of B^{l} to c_{0}(Z_{0+}, k) has operator q-norm equal to 1 with respect to the supremum q-norm on c_{0}(Z_{0+}, k). This uses (47.3) and (47.4), with r = ∞.

Let X be a nonempty set, let Y be a nonempty subset of X, and let P_{Y} be the projection operator defined in (45.5). If f ∈ ℓ^{r}(X, k) for some r > 0, then P_{Y} (f) ∈ ℓ^{r}(X, k), and ∥P_{Y} (f)∥_{r} ≤ ∥f∥_{r}, (47.5)

with equality when f is supported in Y. Thus P_{Y} defines a bounded linear mapping from ℓ^{r}(X, k) into itself, with operator q or r-norm equal to 1. Similarly, P_{Y} maps c_{0}(X, k) into itself, with operator q-norm equal to 1 with respect to the supremum q-norm on c_{0}(X, k). These statements about operator norms can also be obtained from the remarks in Section 19, because P_{Y} corresponds to multiplication by a k-valued function on X with supremum q-norm equal to 1, as in Section 45.

Let R_{Y} be the restriction mapping from c(X, k) onto c(Y, k), as in Section 45. If f ∈ ℓ^{r}(X, k) for some r > 0, then R_{Y} (f) ∈ ℓ^{r}(Y, k), and

∥R_{Y} (f)∥_{ℓ^{r}(Y,k)} ≤ ∥f∥_{ℓ^{r}(X,k)}, (47.6)

with equality when f is supported in Y. Hence R_{Y} defines a bounded linear mapping from ℓ^{r}(X, k) into ℓ^{r}(Y, k), with operator q or r-norm equal to 1. Similarly, R_{Y} maps c_{0}(X, k) into c_{0}(Y, k), with operator q-norm equal to 1 with respect to the corresponding supremum q-norms. Let E_{Y} be the extension mapping from c(Y, k) into c(X, k), as in (45.1). If f_{0} ∈ ℓ^{r}(Y, k) for some r > 0, then E_{Y} (f_{0}) ∈ ℓ^{r}(X, k), and

∥E_{Y} (f_{0})∥_{ℓ^{r}(X,k)} = ∥f_{0}∥_{ℓ^{r}(Y,k)}, (47.7)

so that E_{Y} defines an isometric linear mapping from ℓ^{r}(Y, k) into ℓ^{r}(X, k). Note that E_{Y} maps c_{0}(Y, k) into c_{0}(X, k) too. It is easy to see that R_{Y} maps ℓ^{r}(X, k) onto ℓ^{r}(Y, k) for every r > 0, and that R_{Y} maps c_{0}(X, k) onto c_{0}(Y, k), using (45.4).

48 Adjoint mappings

In this section, we take k = R or C, with the standard absolute value function. Remember that the standard inner products on ℓ^{2}(Z_{0+}, R) and ℓ^{2}(Z_{0+}, C) are given by

⟨f, g⟩ = ∑_{j=0}^{∞} f(j) g(j) (48.1)

and

⟨f, g⟩ = ∑_{j=0}^{∞} f(j) \overline{g(j)}, (48.2)

as in (11.3) and (11.4), respectively. The forward and backward shift operators A, B from Section 43 define bounded linear mappings from ℓ^{2}(Z_{0+}, R) and ℓ^{2}(Z_{0+}, C) into themselves, as in the previous section. If f, g are squaresummable real or complex-valued functions on Z_{0+}, then one can check that

⟨A(f), g⟩ = ⟨f, B(g)⟩. (48.3)

This is analogous to (46.4). It follows that

A^{∗} = B (48.4)

as bounded linear mappings on ℓ^{2}(Z_{0+}, R) or ℓ^{2}(Z_{0+}, C), where A^{∗} is the adjoint of A, as in Section 21. Of course,

B^{∗} = A, (48.5)

for essentially the same reasons, or by taking the adjoints of both sides of (48.4).

Let X be a nonempty set, and let Y be a nonempty subset of X. As before, the standard inner products on ℓ^{2}(X, R) and ℓ^{2}(X, C) are given by

⟨f, g⟩_{X} = ∑_{x∈X} f(x) g(x) (48.6)

and

⟨f, g⟩_{X} = ∑_{x∈X} f(x) \overline{g(x)}, (48.7)

respectively. The projection mapping P_{Y} in (45.5) determines a bounded linear mapping from each of ℓ^{2}(X, R) and ℓ^{2}(X, C) into itself, as in the previous section. If f, g are square-summable real or complex-valued functions on X, then ⟨P_{Y} (f), g⟩_{X} = ⟨f, P_{Y} (g)⟩_{X}, (48.8)

as in (46.7). This means that P_{Y}^{∗} = P_{Y} (48.9)

as bounded linear mappings on ℓ^{2}(X, R) or ℓ^{2}(X, C).

The standard inner products on ℓ^{2}(Y, R) and ℓ^{2}(Y, C) are given by

⟨f_{0}, g_{0}⟩_{Y} = ∑_{x∈X} f_{0}(x) g_{0}(x) (48.10)

and

⟨f_{0}, g_{0}⟩_{Y} = ∑_{x∈X} f_{0}(x) \overline{g_{0}(x)}, (48.11)

respectively. As in the previous section, the restriction mapping R_{Y} from Section 45 defines a bounded linear mapping from ℓ^{2}(X, k) into ℓ^{2}(Y, k) for k = R, C. Similarly, the extension mapping E_{Y} in (45.1) defines a bounded linear mapping from ℓ^{2}(Y, k) into ℓ^{2}(X, k) for k = R, C. If f is a square-summable real or complex-valued function on X, and g_{0} is a square-summable real or complex-valued function on Y, then

⟨R_{Y} (f), g_{0}⟩_{Y} = ⟨f, E_{Y} (g_{0})⟩_{X}, (48.12)

as in (46.8). This implies that

R_{Y}^{∗} = E_{Y} (48.13)

as bounded linear mappings from ℓ^{2}(Y, k) into ℓ^{2}(X, k), for k = R, C. We also have that E_{Y}^{∗} = R_{Y} (48.14)

as bounded linear mappings from ℓ^{2}(X, k) into ℓ^{2}(Y, k), for k = R, C. This can be obtained in essentially the same way, or by taking adjoints of both sides of (48.13).

49 Eigenfunctions for unilateral shifts

Let k be a field, and let A and B be the forward and backward shift operators on c(Z_{0+}, k), as in Section 43. One can check that A has no nontrivial eigenvectors in c(Z_{0+}, k), so that A has no eigenvalues in k as a linear mapping from c(Z_{0+}, k) into itself. More precisely, 0 is not an eigenvalue of A, because A is injective on c(Z_{0+}, k). If a ∈ k, a ≠ 0, f ∈ c(Z_{0+}, k), and A(f) = a f, then one can verify that f ≡ 0 on Z_{0+}, using the definition of A. This corresponds to the fact that the bilateral shift operator T on c(Z, k) has no nontrivial eigenfunctions supported in Z_{0+}, as in Section 26.

Let a ∈ k be given, and let e_{a,0} be the k-valued function defined on Z_{0+} by

e_{a,0}(j) = a^{j} (49.1)

for every j ≥ 0. This is interpreted as being equal to 1 when j = 0, as usual, even when a = 0. If a ≠ 0, then e_{a,0} is the same as the restriction to Z_{0+} of the function e_{a} defined on Z in (26.2). Observe that

(B(e_{a,0}))(j) = e_{a,0}(j + 1) = a^{j+1} = a e_{a,0}(j) (49.2) for every a ∈ k and j ≥ 0, so that

B(e_{a,0}) = a e_{a,0} (49.3)

as elements of c(Z_{0+}, k). Thus e_{a,0} is an eigenvector for B on c(Z_{0+}, k) with eigenvalue a for each a ∈ k. It is easy to see that every eigenvector of B on c(Z_{0+}) with eigenvalue a is a multiple of e_{a,0}. Note that e_{a,0} has finite support in Z_{0+} only when a = 0, so that 0 is the only eigenvalue of B on c_{00}(Z_{0+}, k).

If a ∈ k, a ≠ 0, and f ∈ c(Z_{0+}, k), then

(A(e_{a,0} f))(j) = a^{j−1} (A(f))(j) = a^{−1} e_{a,0}(j) (A(f))(j) (49.4)

for every j ≥ 0. Of course, each of these three expressions is equal to 0 when j = 0, by the definition (43.1) of A. Thus

A(e_{a,0} f) = a^{−1} e_{a,0} A(f) (49.5)

as k-valued functions on Z_{0+}. Let M_{e_{a,0}} be the multiplication operator on c(Z_{0+}, k) corresponding to e_{a,0}, as in (19.1). Using this, (49.5) can be reformulated as saying that A ∘ M_{e_{a,0}} = a^{−1} M_{e_{a,0}} ∘ A (49.6)

as linear mappings from c(Z_{0+}, k) into itself. Note that (49.1) is nonzero for every j ∈ Z when a ≠ 0, in which case 1/e_{a,0} = e_{1/a,0} and M_{e_{a,0}}^{−1} = M_{1/e_{a,0}} = M_{e_{1/a,0}}. It is easy to see that

A ∘ M_{e_{a,0}}^{−1} = a M_{e_{a,0}}^{−1} ∘ A (49.7)

for every a ∈ k \ {0}, by rearranging the operators in (49.6), or by applying (49.6) to 1/a. It follows that

M_{e_{a,0}} ∘ A ∘ M_{e_{a,0}}^{−1} = a A (49.8)

for every a ∈ k \ {0}, as linear mappings from c(Z_{0+}, k) into itself.

If f, g ∈ c(Z_{0+}, k), then

B(f g) = B(f) B(g) (49.9)

as k-valued functions on Z_{0+}, by the definition (43.2) of B. In particular,

B(e_{a,0} f) = B(e_{a,0}) B(f) = a e_{a,0} B(f) (49.10)

for every a ∈ k and f ∈ c(Z_{0+}, k), using (49.3) in the second step. This implies that B ∘ M_{e_{a,0}} = a M_{e_{a,0}} ∘ B (49.11)

for every a ∈ k, as linear mappings from c(Z_{0+}, k) into itself. If a ≠ 0, then we have that B ∘ M_{e_{a,0}}^{−1} = a^{−1} M_{e_{a,0}}^{−1} ∘ B, (49.12)

by rearranging the operators in (49.11), or applying the previous statement to 1/a. Thus M_{e_{a,0}} ∘ B ∘ M_{e_{a,0}}^{−1} = a^{−1} B (49.13)

as linear mappings on c(Z_{0+}, k) when a ≠ 0.

50 Eigenvalues of unilateral shifts

Let k be a field, and let | · | be a q-absolute value function on k for some q > 0. Also let a ∈ k be given, and let e_{a,0} be defined on Z_{0+} as in (49.1). Observe that (50.1) e_{a,0} ∈ ℓ^{∞}(Z_{0+}, k) if and only if |a| ≤ 1.

Similarly, (50.2) e_{a,0} ∈ c_{0}(Z_{0+}, k) if and only if |a| < 1.

If 0 < r < ∞, then

(50.3) e_{a,0} ∈ ℓ^{r}(Z_{0+}, k) if and only if |a| < 1.

Let A, B be the usual forward and backward shift operators on c(Z_{0+}, k), as in Section 43. Remember that the restrictions of A and B to ℓ^{r}(Z_{0+}, k) define bounded linear mappings from ℓ^{r}(Z_{0+}, k) into itself for every r > 0, as in Section 47. It follows from (50.1) and the remarks about eigenfunctions of B on c(Z_{0+}, k) in the previous section that a ∈ k is an eigenvalue of B on ℓ^{∞}(Z_{0+}, k) if and only if |a| ≤ 1. Using (50.2), we get that a ∈ k is an eigenvalue of B on c_{0}(Z_{0+}, k) if and only if |a| < 1. If 0 < r < ∞, then a ∈ k is an eigenvalue of B on ℓ^{r}(Z_{0+}, k) if and only if |a| < 1, by (50.3).

Let us now consider approximate eigenvalues of A and B on ℓ^{r}(Z_{0+}, k) and c_{0}(Z_{0+}, k). Of course, if a ∈ k is an approximate eigenvalue of A or B on c_{0}(Z_{+}, k) with respect to the supremum norm, then a is an approximate eigenvalue of A or B, respectively, on ℓ^{∞}(Z_{0+}, k). Remember that A defines an isometric linear mapping from ℓ^{r}(Z_{0+}, k) into itself for every r > 0, as in Section 47. If a ∈ k is an approximate eigenvalue of A on ℓ^{r}(Z_{0+}, k) for some r > 0, then it follows that |a| = 1, as in Section 30. Similarly, the restriction of B to ℓ^{r}(Z_{0+}, k) has operator q or r-norm equal to 1, as appropriate, as in Section 47. If a ∈ k is an approximate eigenvalue of B on ℓ^{r}(Z_{0+}, k) for some r > 0, then |a| ≤ 1, as in (30.14). We shall restrict our attention to |a| = 1 for B as well, since the case where |a| < 1 is covered by the remarks in the preceding paragraph.

Let a ∈ k with |a| = 1 be given, and let r be a positive real number. Also let T be the forward shift operator on c(Z, k), as in Section 3, and remember that A basically corresponds to the restriction of T to k-valued functions on Z that are supported in Z_{0+}. We have seen that a is an approximate eigenvalue for T on ℓ^{r}(Z, k), as in Section 32. The same type of construction can be used to show that a is an approximate eigenvalue for A on ℓ^{r}(Z_{0+}, k). More precisely, one can use functions defined as in (32.1), with j_{1} ≥ 0, restricted to Z_{0+}.

There are a few minor differences between the analogous argument for B and the previous situation. Let j_{0} be a nonnegative integer, and let f_{0} be the k-valued function defined on Z_{0+} by

(50.4)  f_{0}(j) = a^{j} when 0 ≤ j ≤ j_{0}
= 0 when j ≥ j_{0} + 1.

This corresponds to (32.1), restricted to Z_{0+}, with j_{1} = 0, j_{2} = j_{0}, and a replaced with 1/a. Using the definition (43.2) of B, we get that

(50.5)  (B(f_{0}))(j) = f_{0}(j + 1) = a^{j+1} when j ≤ j_{0} − 1
= 0  when j ≥ j_{0}.

Thus

(50.6)  a f_{0}(j) − (B(f_{0}))(j) = a^{j_{0}+1} when j = j_{0}
= 0  otherwise,

so that ∥a f_{0} − B(f_{0})∥_{r} = 1. (50.7)

We also have that ∥f_{0}∥_{r} = (j_{0} + 1)^{1/r}, (50.8)

which implies that a is an approximate eigenvalue for B on ℓ^{r}(Z_{0+}, k). Of course, B basically corresponds to T^{−1} in the earlier discussion, and the restriction to Z_{0+} permits us to avoid an extra term in (50.6).

Suppose for the rest of the section that k = R or C, with the standard absolute value function. Let a be a real or complex number with |a| = 1, and let f_{n} be the real or complex-valued function defined on Z for each nonnegative integer n as in (32.6). In order to get functions that are supported in Z_{0+}, one can take f̃_{n}(j) = f_{n}(j − n), (50.9)

for instance. Using the restrictions of these functions to Z_{0+}, one can show that a is an approximate eigenvalue for A on ℓ^{r}(Z_{0+}, k) when 0 < r < ∞, and on c_{0}(Z_{0+}, k) with the supremum norm. This is basically the same as in Section 32, with additional translations as in (50.9), because A corresponds to the restriction of T to functions supported in Z_{0+}.

As before, there are analogous arguments for B, with some minor differences. In order to use the same type of functions for B, one should replace a with 1/a, because B corresponds to T^{−1}. It is not necessary to use translations as in (50.9), and instead one can simply restrict the functions to Z_{0+}. However, instead of using the same type of functions as in (32.6), one can do the following. Let {a_{n}}_{n=1}^{∞} be a sequence of real or complex numbers, as appropriate, that converges to a, and satisfies |a_{n}| < 1 for each n. If e_{a_{n},0} is defined on Z_{0+} as in (49.1), then e_{a_{n},0} is an eigenfunction for B with eigenvalue a_{n} for each n, as in (49.3). We also have that e_{a_{n},0} ∈ c_{0}(Z_{0+}, k) for each n, by (50.2), and that e_{a_{n},0} ∈ ℓ^{r}(Z_{0+}, k) for every positive real number r and n ≥ 1, by (50.3). Using this, one can check that a is an approximate eigenvalue of B on ℓ^{r}(Z_{0+}, k) when 0 < r < ∞, and on c_{0}(Z_{0+}, k) with respect to the supremum norm.

51 The case where q = r = ∞

Let k be a field with an ultrametric absolute value function | · |, and let a ∈ k be given, with |a| = 1. If f ∈ c(Z_{0+}, k) and A(f) is as in (43.1), then

(51.1)  a f(j) − (A(f))(j) = a f(j) − f(j − 1) when j ≥ 1
= a f(0)  when j = 0.

Let l be a nonnegative integer, and observe that

a^{l} f(l) = ∑_{j=1}^{l}(a^{j} f(j) − a^{j−1} f(j − 1)) + f(0), (51.2)

where the summation on the right side is interpreted as being equal to 0 when l = 0. Equivalently,

a^{l} f(l) = ∑_{j=0}^{l} a^{j−1} (a f(j) − (A(f))(j)), (51.3)

where the f(0) term on the right side of (51.2) corresponds to the j = 0 in the sum on the right side of (51.3). Thus

|f(l)| ≤ max_{0≤j≤l} |a f(j) − (A(f))(j)|, (51.4)

by the ultrametric version of the triangle inequality.

If a f − A(f) is bounded on Z_{0+}, then it follows that f is bounded on Z_{0+}, with ∥f∥_{∞} ≤ ∥a f − A(f)∥_{∞}. (51.5)

This could also be obtained from (33.5), applied to k-valued functions on Z supported in Z_{0+}. It follows that a is not an approximate eigenvalue of A on ℓ^{∞}(Z_{0+}, k). Note that

∥a f − A(f)∥_{∞} ≤ max(∥a f∥_{∞}, ∥A(f)∥_{∞}) ≤ ∥f∥_{∞} (51.6)

for every f ∈ ℓ^{∞}(Z_{0+}, k), by the ultrametric version of the triangle inequality. Hence ∥a f − A(f)∥_{∞} = ∥f∥_{∞} (51.7)

for every f ∈ ℓ^{∞}(Z_{0+}, k), by (51.5) and (51.6).

Let f ∈ c(Z_{0+}, k) and nonnegative integers j_{1}, j_{2} be given, with j_{1} < j_{2}. Observe that

(51.8) a^{−j_{1}} f(j_{1}) − a^{−j_{2}} f(j_{2}) = ∑_{j=j_{1}}^{j_{2}−1}(a^{−j} f(j) − a^{−j−1} f(j + 1))

= ∑_{j=j_{1}}^{j_{2}−1} a^{−j−1} (a f(j) − (B(f))(j)),

where B(f) is as in (43.2). This implies that

|a^{−j_{1}} f(j_{1}) − a^{−j_{2}} f(j_{2})| ≤ max_{j_{1}≤j≤j_{2}−1} |a f(j) − (B(f))(j)|, (51.9)

by the ultrametric version of the triangle inequality. If a f − B(f) is bounded on Z_{0+}, then we have that

|a^{−j_{1}} f(j_{1}) − a^{−j_{2}} f(j_{2})| ≤ ∥a f − B(f)∥_{∞} (51.10)

for every j_{1}, j_{2} ≥ 0. It follows that f is bounded on Z_{0+}, with

∥f∥_{∞} ≤ max (∥a f − B(f)∥_{∞}, inf_{j≥0} |f(j)|), (51.11)

by the ultrametric version of the triangle inequality.

If f vanishes at infinity on Z_{0+}, then

∥f∥_{∞} ≤ ∥a f − B(f)∥_{∞}, (51.12)

by (51.11). This means that a is not an approximate eigenvalue of B on c_{0}(Z_{0+}, k) with respect to the supremum ultranorm. As before,

∥a f − B(f)∥_{∞} ≤ max(∥a f∥_{∞}, ∥B(f)∥_{∞}) ≤ ∥f∥_{∞} (51.13)

for every f ∈ ℓ^{∞}(Z_{0+}, k), by the ultrametric version of the triangle inequality. Thus ∥a f − B(f)∥_{∞} = ∥f∥_{∞} (51.14)

for every f ∈ c_{0}(Z_{0+}, k). Note that

sup_{j_{1},j_{2}≥0} |a^{−j_{1}} f(j_{1}) − a^{−j_{2}} f(j_{2})| = ∥a f − B(f)∥_{∞} (51.15)

for every f ∈ ℓ^{∞}(Z_{0+}, k), because of (51.10) and the definition of B.

52 Multiplicative inverses in k[[X]]

Let k be a field, and let X be an indeterminate. Of course, elements of k can be identified with formal polynomials in X of degree 0. More precisely, such a constant polynomial is a scalar multiple of X^{0}, which is typically omitted from the notation. This defines a natural embedding of k into k[X]. In particular, the multiplicative identity element 1 in k corresponds to a constant polynomial in X, which may be denoted 1 as well. This is the multiplicative identity element in k[[X]]. Thus F (X) ∈ k[[X]] has a multiplicative inverse in k[[X]] when there is a G(X) ∈ k[[X]] such that

F (X) G(X) = 1. (52.1)

Let a ∈ k and a nonnegative integer n be given, so that ∑_{j=0}^{n} a^{j} X^{j} defines an element of k[X]. As usual,

(1 − a X) ∑_{j=0}^{n} a^{j} X^{j} = 1 − a^{n+1} X^{n+1}. (52.2)

Similarly, ∑_{j=0}^{∞} a^{j} X^{j} defines an element of k[[X]], and

(1 − a X) ∑_{j=0}^{∞} a^{j} X^{j} = 1. (52.3)

Thus 1 − a X is invertible in k[[X]], with

(1 − a X)^{−1} = ∑_{j=0}^{∞} a^{j} X^{j}. (52.4)

Now let a(X) ∈ k[[X]] be given, so that (a(X) X)^{l} = a(X)^{l} X^{l} is defined as a formal power series in X for every nonnegative integer l. If n is a nonnegative integer, then

(1 − a(X) X) ∑_{l=0}^{n} a(X)^{l} X^{l} = 1 − a(x)^{n+1} X^{n+1}, (52.5)

as in (52.2). The sum

∑_{l=0}^{∞} a(X)^{l} X^{l} (52.6)

can be defined as a formal power series in X as well, because the coefficient of X^{j} reduces to a finite sum for each j ≥ 0. One can also check that

(1 − a(X) X) ∑_{l=0}^{∞} a(X)^{l} X^{l} = 1, (52.7)

as in (52.3). This implies that 1 − a(X) X is invertible in k[[X]], with inverse equal to (52.6).

Let F (X) ∈ k[[X]] be given. If F (X) has a multiplicative inverse in k[[X]], then it is easy to see that the constant term in F (X) is not equal to 0. Conversely, if the constant term in F (X) is not equal to 0, then F (X) can be expressed as F (X) = b (1 − a(X) X), (52.8)

where b ∈ k, b ≠ 0, and a(X) ∈ k[[X]]. This implies that F (X) has a multiplicative inverse in k[[X]], by the remarks in the preceding paragraph.

53 Inverting I − a A on c(Z_{0+}, k)

Let k be a field, and let a ∈ k be given. If n is a nonnegative integer, then put

C_{a,n} = ∑_{l=0}^{n} a^{l} A^{l}, (53.1)

where A is as in (43.1). This defines a linear mapping from c(Z_{0+}, k) into itself. Equivalently,

(C_{a,n}(f))(j) = ∑_{l=0}^{n} a^{l} (A^{l}(f))(j) = ∑_{l=0}^{min(j,n)} a^{l} f(j − l) (53.2)

for every f ∈ c(Z_{0+}, k) and j ≥ 0, using (43.6) in the second step. Note that

(I − a A) ∘ C_{a,n} = C_{a,n} ∘ (I − a A) = I − a^{n+1} A^{n+1} (53.3)

as linear mappings on c(Z_{0+}, k) for each n ≥ 0, as in (23.1).

If f ∈ c(Z_{0+}, k) and j ∈ Z_{0+}, then put

(C_{a}(f))(j) = ∑_{l=0}^{j} a^{l}f(j − l). (53.4)

This defines a k-valued function on Z_{0+}, and C_{a} defines a linear mapping from c(Z_{0+}, k) into itself. Comparing (53.4) with (53.2), we get that

(C_{a}(f))(j) = (C_{a,n}(f))(j) (53.5)

when j ≤ n. Equivalently,

(C_{a}(f))(j) = ∑_{l=0}^{∞} a^{l} (A^{l}(f))(j) (53.6)

for every j ≥ 0, where the right side of (53.6) reduces to the finite sum in (53.4), because of (43.6). Basically, C_{a} corresponds to ∑_{l=0}^{∞} a^{l} A^{l}, which is made precise by (53.4) and (53.6).

Let f ∈ c(Z_{0+}, k) be given, and observe that

(C_{a}(a A(f)))(j) = ∑_{l=0}^{∞} a^{l+1} (A^{l+1}(f))(j) (53.7)

for each j ≥ 0, by applying (53.6) to A(f) in place of f. Similarly,

a (A(C_{a}(f)))(j) = ∑_{l=0}^{∞} a^{l+1} (A^{l+1}(f))(j) (53.8)

for each j ≥ 0, by applying A to both sides of (53.6). We also have that

∑_{l=0}^{∞} a^{l+1} (A^{l+1}(f))(j) = ∑_{l=1}^{∞} a^{l} (A^{l}(f))(j) = (C_{a}(f))(j) − f(j) (53.9)

for each j ≥ 0, using (53.6) in the second step. Combining this with (53.7) and (53.8), we get that

aA(C_{a}(f)) = C_{a}(aA(f)) = C_{a}(f) − f. (53.10)

Thus a A ∘ C_{a} = a C_{a} ∘ A = C_{a} − I (53.11)

as linear mappings from c(Z_{0+}, k) into itself, so that

(I − a A) ∘ C_{a} = C_{a} ∘ (I − a A) = I. (53.12)

Of course, this means that I−a A is invertible as a linear mapping from c(Z_{0+}, k) into itself, with (I − a A)^{−1} = C_{a}. (53.13)

If | · | is an ultrametric absolute value function on k and |a| ≤ 1, then it is easy to see that C_{a} maps ℓ^{∞}(Z_{0+}, k) into itself.

If b ∈ k and b ≠ 0, then it follows that b I − A = b (I − b^{−1} A) is invertible as a linear mapping from c(Z_{0+}, k) into itself, with

(b I − A)^{−1} = b^{−1} (I − b^{−1} A)^{−1} = b^{−1} C_{1/b}. (53.14)

We have already seen in Section 49 that A has no eigenvalues in k as a linear mapping from c(Z_{0+}, k) into itself, which is the same as saying that b I − A is injective on c(Z_{0+}, k) for every b ∈ k. If b = 0, then b I − A = −A does not map c(Z_{0+}, k) onto itself, and hence is not invertible on c(Z_{0+}, k).

54 ℓ^{r} Estimates, r ≤ q, r < ∞

Let k be a field with a q-absolute value function | · | for some q > 0, and let r be a positive real number with r ≤ q. Remember that the forward shift operator A in (43.1) defines an isometric linear mapping from ℓ^{r}(Z_{0+}, k) into itself, as in Section 47. Let a ∈ k and n ∈ Z_{0+} be given, and let C_{a,n} be as in (53.1). If f ∈ ℓ^{r}(Z_{0+}, k), then

∥C_{a,n}(f)∥_{r}^{r} = ∥∑_{l=0}^{n} a^{l} A^{l}(f)∥_{r}^{r} ≤ ∑_{l=0}^{n} |a^{l}|^{r} ∥A^{l}(f)∥_{r}^{r} = ( ∑_{l=0}^{n} |a|^{l r}) ∥f∥_{r}^{r}. (54.1)

This uses the fact that ∥ · ∥_{r} defines an r-norm on ℓ^{r}(Z_{0+}, k) when r ≤ q, as in Section 8, in the second step. Thus C_{a,n} defines a bounded linear mapping from ℓ^{r}(Z_{0+}, k) into itself, with operator r-norm less than or equal to

( ∑_{l=0}^{n} |a|^{l r})^{1/r}. (54.2)

If f(j) = 0 for every j ≥ 1, then (A^{l}(f))(j) = 0 when j ≠ l, as in (43.6). In this case, one can check that equality holds in the second step in (54.1). This implies that the operator r-norm of C_{a,n} on ℓ^{r}(Z_{0+}, k) is equal to (54.2).

Let C_{a} be as in (53.4), and let f ∈ c(Z_{0+}, k) be given. Observe that

∑_{j=0}^{n} |(C_{a}(f))(j)|^{r} = ∑_{j=0}^{n} |(C_{a,n}(f))(j)|^{r} (54.3)

for each n ≥ 0, by (53.5). Of course,

∑_{j=0}^{n} |(C_{a,n}(f))(j)|^{r} ≤ ∑_{j=0}^{∞} |(C_{a,n}(f))(j)|^{r} = ∥C_{a,n}(f)∥_{r}^{r} (54.4)

for every n ≥ 0. If f(j) = 0 when j ≥ 1, then (C_{a,n}(f))(j) = 0 when j > n, by (53.2). This implies that equality holds in the first step in (54.4) for every n ≥ 0 in this situation.

Suppose that |a| < 1, so that (54.2) is less than or equal to

( ∑_{l=0}^{∞} |a|^{l r})^{1/r} = (1 − |a|^{r})^{−1/r} (54.5)

for each n ≥ 0. If f ∈ ℓ^{r}(Z_{0+}, k), then it follows that

∑_{j=0}^{n} |(C_{a}(f))(j)|^{r} ≤ (1 − |a|^{r})^{−1} ∥f∥_{r}^{r} (54.6)

for every n ≥ 0, by (54.1), (54.3), and (54.4). This implies that

∥C_{a}(f)∥_{r}^{r} = ∑_{j=0}^{∞} |(C_{a}(f))(j)|^{r} ≤ (1 − |a|^{r})^{−1} ∥f∥_{r}^{r}, (54.7)

and in particular that C_{a}(f) ∈ ℓ^{r}(Z_{0+}, k). Thus C_{a} defines a bounded linear mapping from ℓ^{r}(Z_{0+}, k) into itself when |a| < 1, with operator r-norm less than or equal to (54.5). If f(0) = 1 and f(j) = 0 when j ≥ 1, then

(C_{a}(f))(j) = a^{j} (54.8)

for every j ≥ 0, by (53.4). In this case, ∥C_{a}(f)∥_{r} is equal to (54.5), so that equality holds in the second step in (54.7). Hence the operator r-norm of C_{a} on ℓ^{r}(Z_{0+}, k) is equal to (54.5).

By construction, C_{a,n} is the same as the n partial sum of the infinite series

∑_{l=0}^{∞} a^{l} A^{l} (54.9)

of linear mappings on c(Z_{0+}, k). Of course, the operator r-norm of a^{l} A^{l} on ℓ^{r}(Z_{0+}, k) is equal to |a|^{l} for every l ≥ 0, because A^{l} is an isometry on ℓ^{r}(Z_{0+}, k).

If |a| < 1, then it follows that (54.9) converges r-absolutely as an infinite series of bounded linear mappings on ℓ^{r}(Z_{0+}, k), with respect to the operator r-norm. One can check that C_{a,n} converges to C_{a} as n → ∞ as a sequence of bounded linear mappings on ℓ^{r}(Z_{0+}, k) with respect to the operator r-norm when |a| < 1, using the same type of simple estimates as before. This means that (54.9) converges to C_{a} as an infinite series of bounded linear mappings on ℓ^{r}(Z_{0+}, k). Note that k is not required to be complete with respect to the q-metric associated to | · | here. In this situation, we were able to define C_{a} directly, in terms of finite sums.

55 Estimates for k = R, C

Let us take k = R or C with the standard absolute value function in this section, so that q = 1. We shall also restrict our attention to r ≥ 1, since the case where r ≤ 1 is covered by the discussion in the previous section. As before, the forward shift operator A in (43.1) defines an isometric linear mapping from ℓ^{r}(Z_{0+}, k) into itself for each r. Let a ∈ k and n ∈ Z_{0+} be given again, and let C_{a,n} be as in (53.1). If r ≥ 1 and f ∈ ℓ^{r}(Z_{0+}, k), then

∥C_{a,n}(f)∥_{r} = ∥∑_{l=0}^{n} a^{l} A^{l}(f)∥_{r} ≤ ∑_{l=0}^{n} |a|^{l} ∥A^{l}(f)∥_{r} = ( ∑_{l=0}^{n} |a|^{l}) ∥f∥_{r}. (55.1)

This uses the fact that ∥ · ∥_{r} defines a norm on ℓ^{r}(Z_{0+}, k) when r ≥ 1, as in Section 8. Thus C_{a,n} defines a bounded linear mapping from ℓ^{r}(Z_{0+}, k) into itself when r ≥ 1, with operator norm less than or equal to

∑_{l=0}^{n} |a|^{l}. (55.2)

Let C_{a} be as in (53.4) again, and let f ∈ c(Z_{0+}, k) be given. As in (54.3) and (54.4), we have that

( ∑_{j=0}^{n} |(C_{a}(f))(j))|^{r})^{1/r} = ( ∑_{j=0}^{n} |(C_{a,n}(f))(j)|^{r})^{1/r} ≤ ∥C_{a,n}(f)∥_{r} (55.3)

when r < ∞, and

max_{0≤j≤n} |(C_{a}(f))(j)| = max_{0≤j≤n} |(C_{a,n}(f))(j)| ≤ ∥C_{a,n}(f)∥_{∞}, (55.4)

using (53.5) in the first step of each. Suppose that |a| < 1, so that (55.2) is less than or equal to

∑_{l=0}^{∞} |a|^{l} = (1 − |a|)^{−1} (55.5)

for each n ≥ 0. If r ≥ 1 and f ∈ ℓ^{r}(Z_{0+}, k), then we get that

∥C_{a}(f)∥_{r} ≤ (1 − |a|)^{−1} ∥f∥_{r}, (55.6) using (55.1), (55.3), and (55.4). This shows that C_{a} defines a bounded linear mapping from ℓ^{r}(Z_{0+}, k) into itself when |a| < 1 and r ≥ 1, with operator norm less than or equal to (55.5).

As before, C_{a} basically corresponds to the infinite series

∑_{l=0}^{∞} a^{l} A^{l}. (55.7)

If |a| < 1, then this series converges absolutely as an infinite series of bounded linear mappings on ℓ^{r}(Z_{0+}, k) with respect to the corresponding operator norm, as in Section 23. One can also check that C_{a} maps c_{0}(Z_{0+}, k) into itself when |a| < 1. This can be verified directly from the original definition of C_{a}, or by considering (55.7) as an absolutely convergent series of bounded linear mappings on c_{0}(Z_{0+}, k) with respect to the supremum norm. This is basically the same as approximating C_{a} by C_{a,n}, and using the fact that C_{a,n} maps c_{0}(Z_{0+}, k) into itself for every n ≥ 0.

As in Section 42, one can show that the operator norm of C_{a} on ℓ^{r}(Z_{0+}, k) is equal to (55.5) for every r ≥ 1 when |a| < 1. Similarly, the operator norm of C_{a} on c_{0}(Z_{0+}, k) with respect to the supremum norm is equal to (55.5) when |a| < 1. Of course, C_{a} is the same as the inverse of I −a A on these spaces when |a| < 1. If b ∈ k and |b| = 1, then b is an approximate eigenvalue of A on these spaces, as in Section 50. This permits one to show that the operator norm of C_{a} on these spaces is greater than or equal to (55.5), in the same way as before.

56 Convergent power series

Let k be a field with a q-absolute value function | · | for some q > 0, and suppose that k is complete with respect to the associated q-metric. Let f ∈ ℓ^{∞}(Z_{0+}, k) and x ∈ k be given, with |x| < 1. Under these conditions,

∑_{j=0}^{∞} f(j) x^{j} (56.1)

converges in k, by the remarks in Section 12. More precisely, if q < ∞, then (56.1) converges q-absolutely, by comparison with the convergent geometric series ∑_{j=0}^{∞} |x|^{q j}. If q = ∞, then it suffices to observe that the terms of the series converge to 0.

Let F (x) denote the value of the sum (56.1). Put g = A(f), where A is the forward shift operator defined in Section 43, as usual. Thus g ∈ ℓ^{∞}(Z_{0+}, k), as in Section 47, so that G(x) can be defined as the sum of the corresponding series. In fact,

G(x) = ∑_{j=0}^{∞}(A(f))(j) x^{j} = ∑_{j=1}^{∞} f(j − 1) x^{j} = ∑_{j=0}^{∞} f(j) x^{j+1} = x F (x), (56.2)

as in (44.3).

Put λ_{x}(f) = F (x), considered now as a linear functional on ℓ^{∞}(Z_{0+}, k). Using (56.2), we get that λ_{x}(A(f)) = x λ_{x}(f) (56.3)

for every f ∈ ℓ^{∞}(Z_{0+}, k). It is easy to see that λ_{x} defines a bounded linear functional on ℓ^{∞}(Z_{0+}, k). This implies that the restriction of λ_{x} to c_{0}(Z_{0+}, k) is a bounded linear functional with respect to the supremum q-norm. Similarly, the restriction of λ_{x} to ℓ^{r}(Z_{0+}, k) is a bounded linear functional for every positive real number r. Note that λ_{x} is not identically 0 on ℓ^{r}(Z_{0+}, k) for any r > 0, or on c_{0}(Z_{0+}, k). It follows from (56.3) that x I − A maps ℓ^{r}(Z_{0+}, k) into the kernel of the restriction of λ_{x} to ℓ^{r}(Z_{0+}, k) for each r > 0, and that x I −A maps c_{0}(Z_{0+}, k) into the kernel of the restriction of λ_{x} to c_{0}(Z_{0+}, k). In particular, x I − A is not surjective on these spaces, and hence not invertible.

Of course, if f ∈ c_{00}(Z_{0+}, k), then (56.1) reduces to a finite sum in k for every x ∈ k, and the value of the sum can be defined without asking k to be complete. If g = A(f), then g ∈ c_{00}(Z_{0+}, k) too, and (56.2) holds for every x ∈ k. Thus λ_{x}(f) = F (x) defines a linear functional on c_{00}(Z_{0+}, k) for every x ∈ k, and satisfies (56.3) for every f ∈ c_{00}(Z_{0+}, k) and x ∈ k. This means that x I − A maps c_{00}(Z_{0+}, k) into the kernel of λ_{x} on c_{00}(Z_{0+}, k) for every x ∈ k. As before, λ_{x} is not identically 0 on c_{00}(Z_{0+}, k) for any x ∈ k, and so x I − A is not surjective on c_{00}(Z_{0+}, k) for any x ∈ k.

57 Hardy spaces

In this section, we take k = C, with the standard absolute value function. Let f be a complex-valued function on Z_{0+} such that

∑_{j=0}^{∞} |f(j)| ρ^{j} (57.1)

converges for every nonnegative real number ρ < 1. The convergence of (57.1) implies that |f(j)| ρ^{j} → 0 as j → ∞, and hence that |f(j)| ρ^{j} is bounded on Z_{0+}. If |f(j)| ρ_{1}^{j} (57.2)

is bounded on Z_{0+} for some ρ_{1} > 0, then (57.1) converges when 0 ≤ ρ < ρ_{1}, by comparison with a convergent geometric series. Thus (57.1) converges for every 0 ≤ ρ < 1 if and only if (57.2) is bounded on Z_{0+} for every 0 ≤ ρ_{1} < 1.

Under these conditions,

F (z) = ∑_{j=0}^{∞} f(j) z^{j} (57.3)

defines a holomorphic function on the open unit disk

U = {z ∈ C : |z| < 1} (57.4) in the complex plane. It is well known that every holomorphic function on U can be expressed in this way. If A(f) is the complex-valued function defined on Z_{0+} as in (43.1), then it is easy to see that A(f) satisfies the same type of conditions as in the preceding paragraph. Thus A(f) also determines a holomorphic function on U, which is the same as z F (z), as in (56.2).

Let 0 ≤ ρ < 1 be given, so that F (ρ z) defines a continuous complex-valued function on the unit circle T. If r is a positive real number, then

(\frac{1}{2π} ∫_{T} |F (ρ z)|^{r} |dz|)^{1/r} (57.5)

can be defined using a Riemann integral with respect to the element |dz| of arclength on T. The analogue of this for r = ∞ is

sup_{z∈T} |F (ρ z)|, (57.6)

where the supremum is attained because T is compact. This is the same as

sup{|F (w)| : w ∈ U, |w| ≤ ρ}, (57.7)

by the maximum principle. Clearly (57.7) increases monotonically in ρ, which means that (57.6) increases monotonically in ρ. It is well known that (57.5) aldo increases monotonically as a function of ρ for each r ∈ R_{+}, because F is holomorphic on U. There are analogous statements for harmonic functions when r ≥ 1.

If 0 < r < ∞, then we put

∥F ∥_{H^{r}} = sup_{0≤ρ<1} (\frac{1}{2π} ∫_{T} |F (ρ z)|^{r} |dz|)^{1/r}, (57.8)

where the supremum is defined as a nonnegative extended real number. The Hardy space H^{r} is defined to be the space of holomorphic functions F on U such that (57.8) is finite, which means that (57.5) is bounded. This is a linear subspace of the space of all holomorphic functions on U. If r ≥ 1, then (57.8) defines a norm on H^{r}, and (57.8) defines an r-norm on H^{r} when 0 < r ≤ 1. Equivalently,

∥F ∥_{H^{r}} = lim_{ρ→1−} (\frac{1}{2π} ∫_{T} |F (ρ z)|^{r} |dz|)^{1/r}, (57.9)

because (57.5) increases monotonically in ρ.

Similarly, H^{∞} is defined to be the space of bounded holomorphic functions on U. This is a subalgebra of the algebra of all holomorphic functions on U. The H^{∞} norm

∥F ∥_{H^{∞}} = sup_{z∈U} |F (z)| (57.10)

is the same as the supremum norm on U, which can also be expressed as

∥F ∥_{H^{∞}} = lim_{ρ→1−} ( sup_{z∈T} |F (ρ z)|), (57.11)

because of the monotonicity of (57.6).

It is well known that (57.5) increases monotonically as a function of r for each ρ ∈ [0, 1), by the inequalities of Jensen or Hölder. It is easy to see directly that (57.5) is less than or equal to (57.6) for every r ∈ R_{+} and ρ ∈ [0, 1). If 0 < r_{1} ≤ r_{2} ≤ ∞, then it follows that

H^{r_{2}} ⊆ H^{r_{1}}, (57.12)

with ∥F ∥_{H^{r_{1}}} ≤ ∥F ∥_{H^{r_{2}}} (57.13)

for every F ∈ H^{r_{2}}. If F ∈ H^{r} for some r > 0, then one can check that G(z) = z F (z) defines an element of H^{r} as well, with

∥G∥_{H^{r}} = ∥F ∥_{H^{r}}. (57.14)

This uses (57.9) when r < ∞, and (57.11) when r = ∞.

If F is given as in (57.3), then

\frac{1}{2π} ∫_{T} |F (ρ z)|^{2} |dz| = ∑_{j=0}^{∞} |f(j)|^{2} ρ^{2j} (57.15)

for every 0 ≤ ρ < 1. This follows from the orthonormality of the z^{j}’s with respect to the usual integral inner product (16.10) for L^{2}(T). Using this, we get that F ∈ H^{2} if and only if f ∈ ℓ^{2}(Z_{0+}, C), with

∥F ∥_{H^{2}} = ( ∑_{j=0}^{∞} |f(j)|^{2})^{1/2} = ∥f∥_{ℓ^{2}(Z_{0+},C)}. (57.16)

Note that a complex-valued function f on Z_{0+} satisfies the conditions mentioned at the beginning of the section when f is bounded on Z_{0+}, and in particular when f ∈ ℓ^{2}(Z_{0+}, k).

If F ∈ H^{r} for some r > 0, then it is well known that the limit of F (ρ z) as ρ → 1− exists for almost every z ∈ T with respect to Lebesgue measure. More precisely, nontangential versions of this limit exist almost everywhere on T. If r < ∞, then F (ρ z) converges to the pointwise limit as ρ → 1− with respect to the L^{r} norm on T. In particular, this implies that ∥F ∥_{H^{r}} is equal to the L^{r} norm of the boundary value function with respect to normalized arclength measure on T. If r = ∞, then ∥F ∥_{H^{∞}} is equal to the L^{∞} norm of the boundary value function.

If F ∈ H^{r} for some r > 0, then the product of F with a bounded holomorphic function on U is in H^{r} too. This defines another type of multiplication operator in this situation.

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[24] E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.

[25] E. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.

[26] E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.

[27] E. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.

[28] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.

Index

absolute convergence, 24 absolute value functions, 12

archimedian, 14 discrete, 15 equivalent, 14 non-archimedian, 14 ultrametric, 13

algebraic dual space, 4

BL(V), 19 BL(V, W), 19 bounded linear functionals, 21 bounded linear mappings, 18 bounded partial sums, 69

C, 13 c(X, k), 5 c_{0}(X, k), 18 c_{00}(X, k), 6 closed balls, 11 convergence of infinite series, 24

discrete metric, 11 dual linear mappings, 4, 37 dual norms, 21, 37 dual spaces, 37

eigenvalues, 46

approximate, 56 eigenvectors, 46 essential range, 59

formal Laurent series, 82 formal polynomials, 82 formal power series, 81 Fourier coefficients, 28, 30

H^{∞}, 101 H^{r}, 101 Hardy spaces, 101 Hilbert spaces, 23

I = I_{V}, 4 indicator functions, 59 invertible bounded linear mappings, 20 isometric linear mappings, 21

k[[X]], 81 k[X], 82

L(V), 4 L(V, W), 4 ℓ^{r}(X, k), 17 linear functionals, 4

metrics, 10 Minkowski’s inequality, 10 multiplication operators, 36, 37

norms, 16

open balls, 11 open sets, 11 Ostrowski’s theorems, 14, 15

p-adic absolute value, 14 p-adic numbers, 14

Q, 14 Q_{p}, 14 q-absolute value functions, 12 q-metrics, 10 q-norms, 16 q-semimetrics, 10 q-seminorms, 15

R, 13 R_{+}, 15 r-summable functions, 17

semi-ultrametrics, 11 semi-ultranorms, 16 semimetrics, 10 seminorms, 16 shift operators,

bilateral, backward, 7 forward, 7 unilateral,

backward, 80 forward, 79

summable functions, 17, 21 supports of functions, 5

T′, 37 T^{alg}, 4 trivial absolute value function, 13 trivial ultranorm, 16

ultrametrics, 11 ultranorms, 16 unitary linear mappings, 23

V′, 37 V^{alg}, 4 vanishing at infinity, 18

Z, 7 Z_{+}, 8 Z_{0+}, 79