INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS

MARIA GIRARDI AND LUTZ WEIS

Abstract. Under fairly mild measurability and integrability conditions on operator-valued kernels, boundedness results for integral operators on Bochner spaces L_{p} (X) are given. In particular, these results are applied to convolutions operators.

1. INTRODUCTION

One of the most commonly used boundedness criterion for integral operators states that, for 1 ≤ p ≤ ∞ and σ-finite measure spaces (T, Σ_{T} , μ) and (S, Σ_{S}, ν), a measurable kernel k: T ×S → ℂ defines a bounded linear operator

K: L_{p} (S, ℂ) → L_{p} (T, ℂ)  via  (Kf) (·) := ∫_{S} k (·, s) f (s) dν (s)

provided

sup_{s∈S} ∫_{T} |k (t, s)| dμ (t) ≤ C  and  sup_{t∈T} ∫_{S} |k (t, s)| dν(s) ≤ C  (1.1)

(see, e.g. [5, Theorem 6.18]). In the theory of evolution equations one frequently uses operatorvalued analogs of this situation, where the kernel k maps T × S into the space B (X, Y) of bounded linear operators from a Banach space X into a Banach space Y and then one desires the boundedness of the corresponding integral operator

K: L_{p} (S, X) → L_{p} (T, Y) .

Such integral operators appear, for example, in solution formulas for inhomogeneous Cauchy problems (see, e.g. [10]) and for Volterra integral equations (see, e.g. [11]) as well as in control theory (see, e.g. [2]); furthermore, the stability of such solutions is often expressed in terms of the boundedness of these operators.

However, difficulties can easily arise since in many situations the kernel k is not measurable with respect to the operator norm because the range of k is not (essentially) valued in a separable subspace of B (X, Y). This paper presents boundedness results for integral operators with operatorvalued kernels under relatively mild measurability and integrability conditions on the kernels.

Date: 4 August 2003.

2000 Mathematics Subject Classification. Primary 45P05, 47G10, 46E40. Secondary 46B20.

Key words and phrases. integral operators on Bochner spaces, operator-valued kernels, convolution with operatorvalued kernels.

Girardi gratefully acknowledges the support of the National Science Foundation (DMS-0306750) and the Alexander von Humboldt Foundation. Weis gratefully acknowledges the support of the Volkswagenstiftung I/78593.

2 GIRARDI AND WEIS

The first step is to place a mild measurability condition on a kernel k: T × S → B (X, Y) to guarantee that if f is in the space E (S, X) of finitely-valued finitely-supported measurable functions then the Bochner integrals

(Kf) (·) := ∫_{S} k (·, s) [f (s)] dν (s)  (1.2)

define a measurable function from T into Y, thus defining a mapping

K: E (S, X) → L_{0} (T, Y) .

Then, to ensure that K linearly extends to a desired superspace, one adds integability conditions, which replace (1.1) in the scalar case and, roughly speaking, take the form

sup_{s∈S} ∫_{T} ∥k (t, s) x∥_{Y} dμ (t) ≤ C ∥x∥_{X}  for each x ∈ X  (1.3)
sup_{t∈T} ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν(s) ≤ C ∥y^{∗}∥_{Y^{∗}}  for each y^{∗} ∈ Y^{∗}  (1.4)

along with appropriate measurability conditions (see Section 3 for the precise formulations). Assume k has the appropriate measurability conditions. Theorem 3.4 shows that if k satisfies (1.3) then K extends to a bounded linear operator from L_{1} (S, X) into L_{1} (T, Y); Theorem 3.6 shows that if k satisfies (1.4) then K extends to a bounded linear operator from the closure of E (S, X) in the L_{∞}-norm into L_{∞} (T, Y). Then Theorem 3.8 uses an interpolation argument to show that if k satisfies (1.3) and (1.4) then K extends to a bounded linear operator from L_{p} (S, X) into L_{p} (T, Y) for 1 < p < ∞. The case p = ∞ is more delicate since E (S, X) is not necessarily dense in L_{∞} (S, X). Theorem 3.11 shows that if k satisfies (1.4) then K can be extended to a bounded linear operator from L_{∞} (S, X) into the space of w^{∗}-measurable μ-essentially bounded functions from T into Y^{∗∗} where the integrals in (1.2) exists (a.e) as Dunford integrals for each f ∈ L_{∞} (S, X); also, sufficient conditions are given to guarantee that K maps L_{∞} (S, X) into L_{∞} (T, Y). Using ideas from the Geometry of Banach Spaces, Example 3.13 shows that, without further assumptions, it is necessary to pass to Y^{∗∗} in Theorem 3.11.

As an immediate consequence of these results, Corollary 5.1 gives boundedness results for convolution operators with operator-valued kernels. A similar result, which inspired this paper, was used to obtain operator-valued Fourier multiplier results [8, 7].

2. NOTATION and BASICS

Throughout this paper, X, Y, and Z are Banach spaces over the field 𝕂 of ℝ or ℂ. Also, X^{∗} is the (topological) dual of X and B(X) is the (closed) unit ball of X. The space B (X, Y) of bounded linear operators from X into Y is endowed with the usual uniform operator topology.

A subspace Z of Y^{∗} τ-norms Y, where τ ≥ 1, provided

∥y∥_{Y} ≤ τ sup_{y^{∗}∈B(Z)} |y^{∗} (y)| for each y ∈ Y .

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 3

If Z τ-norms Y, then the natural mapping

j: Y → Z^{∗} given by ⟨ z, jy ⟩ := ⟨ y, z ⟩ for z ∈ Z

is an isomorphic embedding with

\frac{1}{τ} ∥y∥_{Y} ≤ ∥j(y)∥_{Z^{∗}} ≤ ∥y∥_{Y} ,

in which case Y is identified as a subspace of Z^{∗}.

(T, Σ_{T} , μ) and (S, Σ_{S}, ν) are σ-finite (positive) measure spaces;

Σ_{S}^{finite} := {A ∈ Σ_{S}: ν(A) < ∞}
Σ_{S}^{full} := {A ∈ Σ_{S}: ν (S \ A) = 0} ,

with similar notation for the corresponding subsets of Σ_{T}.

E (S, X) is the space of finitely-valued finitely-supported measurable functions from S into X, i.e.

E (S, X) = {∑_{i=1}^{n} x_{i}1_{A_{i}}: x_{i} ∈ X, A_{i} ∈ Σ_{S}^{finite}, n ∈ ℕ} .

Let Γ be a subspace of X^{∗}. A function f: S → X is

• measurable provided there is a sequence (f_{n})_{n=1}^{∞} from E (S, X) so that lim_{n→∞} ∥f(s) − f_{n}(s)∥_{X} = 0 for ν-a.e. s

• σ (X, Γ)-measurable provided ⟨ f (·) , x^{∗} ⟩ : S → 𝕂 is measurable for each x^{∗} ∈ Γ . The following fact will be used (c.f., e.g., [3, Corollary II.1.4]).

Fact 2.1 (Pettis’s Measurability Theorem). A function f: S → X is measurable if and only if (i) f is essentially separably valued (ii) f is σ (X, Γ)-measurable for some subspace Γ of X^{∗} that 1-norms X. □

L_{0} (S, X) is the space of (equivalence classes of) measurable functions from S into X. The Bochner-Lebesgue space L_{p} (S, X), where 1 ≤ p ≤ ∞, is endowed with its usual norm topology. The space L_{∞}^{w^{∗}} (T, Z^{∗}) of μ-essentially bounded σ (Z^{∗}, Z)-measurable functions from T into Z^{∗} is endowed with the μ-essential supremum norm, under which it becomes a Banach space.

E (S, X) is norm dense in L_{p} (S, X) for 1 ≤ p < ∞. Let L_{∞}^{0} (S, X) be the closure of E (S, X) in the L_{∞} (S, X)-norm. If X is infinite-dimensional, then L_{∞}^{0} (S, X) ≠ L_{∞} (S, X) (provided Σ_{S} contains a countable number of pairwise disjoint sets of strictly positive measure). L_{∞}^{0} (S, X) can be described as follows.

Proposition 2.2. Let f ∈ L_{0} (S, X). Then f ∈ L_{∞}^{0} (S, X) if and only if (1) inf {∥f1_{S\A}∥_{L_{∞}(S,X)} : A ∈ Σ_{S}^{finite}} = 0

(2) there is B ∈ Σ_{S}^{full} so that the set {f (s) : s ∈ B} is relatively compact in X.

Proof. It is clear that each f in E (S, X) satisfies conditions (1) and (2) and therefore each f in L_{∞}^{0} (S, X) also satisfies conditions (1) and (2).

4 GIRARDI AND WEIS

Conversely, for ε > 0, conditions (1) and (2) give a set G (:= A ∩ B) ∈ Σ_{S}^{finite} so that

∥f1_{S\G}∥_{L_{∞}(S,X)} < ε and {f (s) : s ∈ G} is relatively compact ;

thus allowing one to find, via a finite covering of the set f (G) by ε-balls, a function f_{ε} ∈ E (S, X), with support in G, so that ∥f − f_{ε}∥_{L_{∞}(S,X)} < ε. □

Lemma 2.3 will help to deal with the fact that E (S, X) is (usually) not norm dense in L_{∞} (S, X).

Lemma 2.3. Let f ∈ L_{∞} (S, X) and ε > 0. There is a sequence {g_{n}}_{n=1}^{∞} from E (S, X) so that

f (s) = ∑_{n=1}^{∞} g_{n} (s)

∑_{n=1}^{∞} ∥g_{n} (s)∥_{X} ≤ (1 + ε) ∥f∥_{L_{∞}(S,X)}

for a.e. s ∈ S.

Proof. Fix a sequence {ε_{j}}_{j=1}^{∞} of positive numbers so that ε_{1} = 1 and ∑_{j=1}^{∞} ε_{j} < 1 + ε. Choose a sequence {f_{j}}_{j=1}^{∞} from E (S, X) so that, for a.e. s ∈ S,

f_{j} (s) → f (s)  as j → ∞
∥f_{j} (s)∥_{X} ≤ ∥f (s)∥_{X}  for each j ∈ ℕ .

Find a sequence {S_{k}}_{k=1}^{∞} of pairwise disjoint sets from Σ_{S}^{finite} so that ν (S \ ∪_{k=1}^{∞}S_{k}) = 0 and, for each S_{k},

f_{j} → f  uniformly on S_{k}
∥f_{j} (s)∥_{X} ≤ ∥f (s)∥_{X}  for each s ∈ S_{k} and j ∈ ℕ .

Hence, on each S_{k}, there is a sequence {g_{j}^{k}}_{j=1}^{∞} from E (S_{k}, X) so that

f (s) = ∑_{j=1}^{∞} g_{j}^{k} (s)  for each s ∈ S_{k}
∥g_{j}^{k}∥_{L_{∞}(S_{k},X)} ≤ ε_{j} ∥f∥_{L_{∞}(S,X)}  for each j ∈ ℕ .

For n ∈ ℕ, let

g_{n} := ∑_{k<n} (g_{n}^{k}1_{S_{k}}) + (∑_{j=1}^{n} g_{j}^{n}) 1_{S_{n}} .

Thus

g_{1} = g_{1}^{1} 1_{S_{1}}
g_{2} = g_{2}^{1} 1_{S_{1}} + (g_{1}^{2} + g_{2}^{2}) 1_{S_{2}}
g_{3} = g_{3}^{1} 1_{S_{1}} + g_{3}^{2} 1_{S_{2}} + (g_{1}^{3} + g_{2}^{3} + g_{3}^{3}) 1_{S_{3}}
g_{4} = g_{4}^{1} 1_{S_{1}} + g_{4}^{2} 1_{S_{2}} + g_{4}^{3}  1_{S_{3}} + (g_{1}^{4} + g_{2}^{4} + g_{3}^{4} + g_{4}^{4}) 1_{S_{4}} .

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 5

Note that if s ∈ S_{k} then

∑_{n=1}^{∞} g_{n} (s) = ∑_{j=1}^{∞} g_{j}^{k} (s)

and, by the triangle inequality,

∑_{n=1}^{∞} ∥g_{n} (s)∥_{X} ≤ ∑_{j=1}^{∞} ∥g_{j}^{k} (s)∥_{X} .

So clearly the g_{n}’s do as they should. □

Let 1 ≤ p ≤ ∞ and \frac{1}{p} + \frac{1}{p′} = 1. There is a natural isometric embedding of L_{p′} (T, Z^{∗}) into [L_{p} (T, Z)]^{∗} given by

⟨ f, g ⟩ := ∫_{T} ⟨ f (t) , g (t) ⟩ dμ (t) for g ∈ L_{p′} (T, Z^{∗}) , f ∈ L_{p} (T, Z) .

There also is a natural isometric embedding of L_{1} (T, Y^{∗}) into [L_{∞}^{w^{∗}} (T, Y^{∗∗})]^{∗}; indeed, for g = ∑_{i=1}^{n} y_{i}^{∗}1_{B_{i}} ∈ L_{1} (T, Y^{∗}) and f ∈ L_{∞}^{w^{∗}} (T, Y^{∗∗}) let

⟨ f, g ⟩ := ∫_{T} ⟨ g (t) , f (t) ⟩ dμ (t) = ∑_{i=1}^{n} ∫_{B_{i}} ⟨ y_{i}^{∗}, f (t) ⟩ dμ (t)

and observe that ∥g∥_{[L_{∞}^{w^{∗}}(T,Y^{∗∗})]}^{∗} = ∥g∥_{L_{1}(T,Y^{∗})}.

For a mapping

k: T × S → B (X, Y)

the mapping

k^{∗}: T × S → B (Y^{∗}, X^{∗})

is defined by k^{∗} (t, s) := [k (t, s)]^{∗}.

Non-numerical subscripts on constants indicate dependency. All other notation and terminology, not otherwise explained, are as in [3, 9].

3. MAIN RESULTS

Several conditions on a kernel k: T × S → B (X, Y) will be considered. The first one is a mild measurability condition.

Definition 3.1. k: T × S → B (X, Y) satisfies condition (C_{0}) provided that for each A ∈ Σ_{S}^{finite} and each x ∈ X

• there is T_{A,x} ∈ Σ_{T}^{full} so that if t ∈ T_{A,x} then the Bochner integral

∫_{A} k (t, s) x dν (s)

exists

• the mapping

T_{A,x} ∋ t → ∫_{A} k (t, s) x dν (s) ∈ Y

defines a measurable function from T into Y. □

6 GIRARDI AND WEIS

Remark 3.2. Let k: T × S → B (X, Y) satisfy condition (C_{0}). Then

for each f ∈ E (S, X) there is T_{f} ∈ Σ_{T}^{full} so that
if t ∈ T_{f} then the Bochner integral (Kf) (t) := ∫_{S} k (t, s) [f (s)] dν (s) exists

(3.1)

and (3.1) defines a linear mapping

K: E (S, X) → L_{0} (T, Y) .  (3.2)

Next integrability conditions on k are added to ensure that the mapping K in (3.2) extends to the desired superspaces. □

Condition (C_{1}) will be used for the L_{1}-case in Theorem 3.4.

Definition 3.3. k: T × S → B (X, Y) satisfies condition (C_{1}) provided there is a constant C_{1} so that for each x ∈ X

• the mapping T × S ∋ (t, s) → ∥k (t, s) x∥_{Y} ∈ ℝ is product measurable

• there is S_{x} ∈ Σ_{S}^{full} so that
∫_{T} ∥k (t, s) x∥_{Y} dμ (t) ≤ C_{1} ∥x∥_{X}

for each s ∈ S_{x}. □

Note that the first condition guarantees that the mapping T ∋ t → ∥k (t, s) x∥_{Y} ∈ ℝ is measurable for ν-a.e. s. Also, the first condition is often satisfied even though the mapping T × S ∋ (t, s) → k (t, s) x ∈ Y may not be product measurable.

The L_{1}-case is a straightforward extension of the scalar-valued situation.

Theorem 3.4. Let k: T × S → B (X, Y) satisfy conditions (C_{0}) and (C_{1}). Then the integral operator

(Kf) (·) := ∫_{S} k (·, s) f (s) dν (s) for f ∈ E (S, X)

extends to a bounded linear operator

K: L_{1} (S, X) → L_{1} (T, Y)

of norm at most the constant C_{1} from Definition 3.3 .

Proof. Fix f = ∑_{i=1}^{n} x_{i}1_{A_{i}} ∈ E (S, X) with the A_{i}’s disjoint. By condition (C_{1}), for each i,

\begin{aligned}∫_{T} ∫_{S} ∥k (t, s) x_{i}1_{A_{i}} (s)∥_{Y} dν (s) dμ (t) & = ∫_{S} [∫_{T} ∥k (t, s) x_{i}∥_{Y} dμ (t)] 1_{A_{i}} (s) dν (s) \\ & ≤ ∫_{S} C_{1} ∥x_{i}∥_{X} 1_{A_{i}} (s) dν (s) \\ & = C_{1} ∥x_{i}∥_{X} ν (A_{i}) . \end{aligned}  (3.3)

Condition (C_{0}) gives that Kf ∈ L_{0} (T, Y) and also, combined with (3.3), that

∥Kf∥_{L_{1}(T,Y)} = ∫_{T} ∥∑_{i=1}^{n} ∫_{S} k (t, s) x_{i}1_{A_{i}} (s) dν (s)∥_{Y} dμ (t)

≤ ∑_{i=1}^{n} ∫_{T} ∫_{S} ∥k (t, s) x_{i}1_{A_{i}} (s)∥_{Y} dν (s) dμ (t)

≤ ∑_{i=1}^{n} C_{1} ∥x_{i}∥_{X} ν (A_{i}) = C_{1} ∥f∥_{L_{1}(S,X)} .

This completes the proof. □

Condition (C_{∞}^{0}) will be used for the L_{∞}^{0}-case in Theorem 3.6.

Definition 3.5. Let Z be a subspace of Y^{∗}. Then k: T × S → B (X, Y) satisfies condition (C_{∞}^{0}), with respect to Z, provided there is a constant C_{∞}^{0} so that for each y^{∗} ∈ Z there is T_{y^{∗}} ∈ Σ_{T}^{full} so that for each t ∈ T_{y^{∗}}

• the mapping S ∋ s → ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∈ ℝ is measurable

• ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s) ≤ C_{∞}^{0} ∥y^{∗}∥_{Y^{∗}} . □

Theorem 3.6. Let Z be a subspace of Y^{∗} that τ-norms Y. Let k: T × S → B (X, Y) satisfy conditions (C_{0}) and (C_{∞}^{0}) with respect to Z. Then the integral operator

(Kf) (·) := ∫_{S} k (·, s) f (s) dν (s) for f ∈ E (S, X)

extends to a bounded linear operator

K: L_{∞}^{0} (S, X) → L_{∞} (T, Y)

of norm at most τ · C_{∞}^{0} where the constant C_{∞}^{0} is from Definition 3.5.

Proof. Fix f ∈ E (S, X). Fix y^{∗} ∈ Z. Find the corresponding sets T_{f}, T_{y^{∗}} ∈ Σ_{T}^{full} from the definitions of conditions (C_{0}) and (C_{∞}^{0}). If t ∈ T_{f} ∩ T_{y^{∗}} then

|⟨ (Kf) (t) , y^{∗} ⟩| = |⟨ ∫_{S} k (t, s) f (s) dν (s) , y^{∗} ⟩|
≤ ∫_{S} |⟨ f (s) , k^{∗} (t, s) y^{∗} ⟩| dν (s)
≤ ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∥f (s)∥_{X} dν (s)
≤ ∥f∥_{L_{∞}(S,X)} C_{∞}^{0} ∥y^{∗}∥_{Y^{∗}} .

Since T_{f} ∩ T_{y^{∗}} ∈ Σ_{T}^{full} and Z τ-norms Y,

∥Kf∥_{L_{∞}(T,Y)} ≤ τ C_{∞}^{0} ∥f∥_{L_{∞}(S,X)} .

This completes the proof. □

8 GIRARDI AND WEIS

Remark 3.7 (on Theorem 3.6). Note that K maps L_{∞}^{0} (S, X) into L_{∞}^{0} (T, Y) provided that for each x ∈ X and A ∈ Σ_{S}^{finite}

T_{f} ∋ t → ∫_{A} k (t, s) x dν (s) ∈ Y  (3.4)

defines a function in L_{∞}^{0} (T, Y). This will be the case, for example, if μ is a Radon measure on a locally compact Hausdorff space T (e.g., T is a Borel subset of ℝ^{N}, endowed with the Lebesgue measure) and (3.4) defines a function in

C_{0} (T, Y) := {g: T → Y | g is continuous and inf {∥g1_{T \B}∥_{L_{∞}} : B is compact} = 0} ;

indeed, conditions (1) and (2) of Proposition 2.2 are then fulfilled. □

Interpolating between Theorems 3.4 and 3.6 gives the L_{p}-case for 1 < p < ∞.

Theorem 3.8. Let Z be a subspace of Y^{∗} that τ-norms Y and 1 < p < ∞. Let k: T ×S → B (X, Y) satisfy conditions (C_{0}), (C_{1}), and (C_{∞}^{0}) with respect to Z. Then the integral operator

(Kf) (·) := ∫_{S} k (·, s) f (s) dν (s) for f ∈ E (S, X)

extends to a bounded linear operator

K: L_{p} (S, X) → L_{p} (T, Y)

of norm at most (C_{1})^{1/p} (τ · C_{∞}^{0})^{1/p′} where the constants C_{1} and C_{∞}^{0} are from Definitions 3.3 and 3.5.

Proof. The proof follows directly from Theorems 3.4 and 3.6 and Lemma 3.9. □

The below interpolation lemma is a slight improvement on [1, Thm. 5.1.2].

Lemma 3.9. Let the linear mapping

K: E (S, X) → L_{1} (T, Y) + L_{∞} (T, Y)

satisfy, for each f ∈ E (S, X),

∥Kf∥_{L_{1}(T,Y)} ≤ c_{1} ∥f∥_{L_{1}(S,X)} < ∞
∥Kf∥_{L_{∞}(T,Y)} ≤ c_{∞} ∥f∥_{L_{∞}(S,X)} < ∞ .

Then, for each 1 < p < ∞, the mapping K extends to a bounded linear operator

K: L_{p} (S, X) → L_{p} (T, Y)

of norm at most (c_{1})^{1/p} (c_{∞})^{1/p′}.

Proof. Fix B ∈ Σ_{T}^{finite} and a finite measurable partition π of B. Let Σ_{0} := σ_{B} (π) be the σ-algebra of subsets of B that is generated by π. Then the linear mapping

K_{0} : E (S, X) → L_{1} (T, Y) + L_{∞} (T, Y) ,

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 9

given by K_{0}f := 𝔼 ((Kf) 1_{B} | Σ_{0}) where 𝔼 (· | Σ_{0}) is the conditional expectation operator relative to Σ_{0}, satisfies

∥K_{0}f∥_{L_{1}(T,Y)} ≤ c_{1} ∥f∥_{L_{1}(S,X)} < ∞
∥K_{0}f∥_{L_{∞}(T,Y)} ≤ c_{∞} ∥f∥_{L_{∞}(S,X)} < ∞

for each f ∈ E (S, X). Furthermore, K_{0} : L_{∞}^{0} (S, X) → L_{∞}^{0} (T, Y). Thus, by [1, Thm. 5.1.2], for each p ∈ (1, ∞), the linear mapping K_{0} extends to a bounded linear operator from L_{p} (S, X) to L_{p} (T, Y) of norm at most (c_{1})^{1/p} (c_{∞})^{1/p′}.

Next, fix f ∈ E (S, X) and p ∈ (1, ∞). By assumption, Kf ∈ L_{1} (T, Y) ∩ L_{∞} (T, Y); thus, Kf ∈ L_{p} (T, Y). Fix B ∈ Σ_{T}^{finite}. Since T is σ-finite, it suffices to show

∥(Kf) 1_{B}∥_{L_{p}(T,Y)} ≤ (c_{1})^{1/p} (c_{∞})^{1/p′} ∥f∥_{L_{p}(S,X)} .  (3.5)

Find a sequence {g_{n}}_{n=1}^{∞} of functions from E (T, Y) that are supported on B and, for μ-a.e. t,

\begin{aligned}g_{n} & → (Kf) (t) 1_{B} (t) \\ ∥g_{n} (t)∥_{Y} & ≤ ∥(Kf) (t) 1_{B} (t)∥_{Y} . \end{aligned}  (3.6)

Let Σ_{n} := σ_{B} (g_{1}, … g_{n}) be the σ-algebra of subsets of B that is generated by {g_{1}, …, g_{n}}. Note that (Kf) 1_{B} is the limit in L_{p} (T, Y) of {g_{n}}_{n=1}^{∞} (by (3.6)) and thus also of {𝔼 ((Kf) 1_{B} | Σ_{n})}_{n=1}^{∞} since

∥(Kf) 1_{B} − 𝔼 ((Kf) 1_{B} | Σ_{n})∥_{L_{p}(T,Y)}
≤ ∥(Kf) 1_{B} − g_{n}∥_{L_{p}(T,Y)} + ∥𝔼 (g_{n} − (Kf) 1_{B} | Σ_{n})∥_{L_{p}(T,Y)} .

But by the previous paragraph, for each n ∈ ℕ,

∥𝔼 ((Kf) 1_{B} | Σ_{n})∥_{L_{p}(T,Y)} ≤ (c_{1})^{1/p} (c_{∞})^{1/p′} ∥f∥_{L_{p}(S,X)} .

Thus (3.5) holds. □

Condition (C_{∞}), a strengthening of condition (C_{∞}^{0}), will be used for the L_{∞}-case in Theorem 3.11.

Definition 3.10. Let Z be a subspace of Y^{∗}. Then k: T × S → B (X, Y) satisfies condition (C_{∞}), with respect to Z, provided there is a constant C_{∞} and T_{0} ∈ Σ_{T}^{full} so that for each t ∈ T_{0} and y^{∗} ∈ Z

• the mapping S ∋ s → k^{∗} (t, s) y^{∗} ∈ X^{∗} is measurable

• ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s) ≤ C_{∞} ∥y^{∗}∥_{Y^{∗}} . □

The L_{∞}-case is more delicate since, in general, E (S, X) is not norm dense in L_{∞} (S, X).

Theorem 3.11. Let Z be a subspace of Y^{∗} that τ-norms Y. Let k: T × S → B (X, Y) satisfy conditions (C_{0}) and (C_{∞}) with respect to Z. In particular, (C_{∞}) gives that for each t ∈ T_{0} ∈ Σ_{T}^{full} and each y^{∗} ∈ Z

∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s) ≤ C_{∞} ∥y^{∗}∥_{Y^{∗}} .

Then the integral operator

K: E (S, X) → L_{∞} (T, Y)

10 GIRARDI AND WEIS

defined by

(Kf) (·) := ∫_{S} k (·, s) f (s) dν (s) for f ∈ E (S, X)  (3.7)

extends (identifying Y as a subspace of Z^{∗}) to a bounded (of norm at most C_{∞}) linear operator

K: L_{∞} (S, X) → L_{∞}^{w^{∗}} (T, Z^{∗})  (3.8)

that is given by

⟨ y^{∗}, (Kf) (t) ⟩ := ∫_{S} ⟨ k (t, s) f (s) , y^{∗} ⟩ dν (s)
for f ∈ L_{∞} (S, X) and t ∈ T_{0} and y^{∗} ∈ Z .

(3.9)

Furthermore, the K of (3.8) maps L_{∞} (S, X) into L_{∞} (T, Y) provided either

(i) Y does not (isomorphically) contain c_{0} or

(ii) for each t ∈ T_{0} the subset {∥k^{∗} (t, ·) y^{∗}∥_{X^{∗}} : y^{∗} ∈ B(Z)} of L_{0} (S, ℝ) is equi-integrable.

Recall that the subset in (ii) is equi-integrable provided if {A_{n}}_{n=1}^{∞} is a sequence from Σ_{S} with A_{n} ⊇ A_{n+1} and ν (∩_{n=1}^{∞}A_{n}) = 0 then

lim_{n→∞} sup_{y^{∗}∈B(Z)} ∫_{A_{n}} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s) = 0 .  (3.10)

Remark 3.12 (on Theorem 3.11).

(a) There can be advantages in taking a proper norming subspace Z ⊊ Y^{∗} over taking Z = Y^{∗}. First, it eases the assumptions of k. Second, Z^{∗} may be much smaller than Y^{∗∗} and so the conclusion K (L_{∞} (S, X)) ⊂ L_{∞}^{w^{∗}} (T, Z^{∗}) may be more useful than K (L_{∞} (S, X)) ⊂ L_{∞}^{w^{∗}} (T, Y^{∗∗}).

For example, if Y = C [0, 1] then Z := L_{1} [0, 1] ⊊ Y^{∗} 1-norms Y; furthermore, Z^{∗} ≃ L_{∞} [0, 1] is nicer than Y^{∗∗} ≃ (M [0, 1])^{∗}, which is very large.

(b) If Y = (Y_{∗})^{∗} is a separable dual space, then Z := Y_{∗} ⊂ Y^{∗} 1-norms Y and, by Pettis’s measurability theorem (Fact 2.1), one has that L_{∞}^{w^{∗}} (T, Z^{∗}) = L_{∞} (T, Y).

(c) If ν (S) < ∞ and for each t ∈ T_{0} there exists q_{t} ∈ (1, ∞] and C_{t} ∈ (0, ∞) such that

sup_{y^{∗}∈B(Z)} ∥k^{∗} (t, ·) y^{∗}∥_{L_{q_{t}}(S,X)} ≤ C_{t} ,

then the equi-integrability condition in (ii) holds; indeed, just apply Hölder’s inequality. (d) Remark 3.7 is valid in this setting also. □

Proof of Theorem 3.11. Fix f ∈ L_{∞} (S, X). Fix t ∈ T_{0}. For each y^{∗} ∈ Z the function

⟨ k (t, ·) f (·) , y^{∗} ⟩ : S → 𝕂

is, by condition (C_{∞}), (a) in L_{0} (S, 𝕂) since ⟨ k (t, ·) f (·) , y^{∗} ⟩ = ⟨ f (·) , k^{∗} (t, ·) y^{∗} ⟩ (b) in L_{1} (S, 𝕂) with

\begin{aligned}∫_{S} |⟨ k (t, s) f (s) , y^{∗} ⟩| dν (s) & ≤ ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∥f (s)∥_{X} dν (s) \\ & ≤ C_{∞} ∥y^{∗}∥_{Z} ∥f∥_{L_{∞}(S,X)} . \end{aligned}  (3.11)

Thus by the Closed Graph Theorem, applied to the mapping

Z ∋ y^{∗} → ⟨ k (t, ·) f (·) , y^{∗} ⟩ ∈ L_{1} (S, 𝕂) ,

the mapping

Z ∋ y^{∗} → ∫_{S} ⟨ k (t, s) f (s) , y^{∗} ⟩ dν (s) ∈ 𝕂

defines an element (Kf) (t) of Z^{∗} that satisfies (3.9).

Let f ∈ E (S, X). By condition (C_{0}), there is T_{f} ∈ Σ_{T}^{full} such that if t ∈ T_{f} then k (t, ·) f (·) ∈ L_{1} (S, Y). For each t ∈ T_{f} ∩ T_{0} ∈ Σ_{T}^{full} and each y^{∗} ∈ Z

⟨ ∫_{S} k (t, s) f (s) dν (s) , y^{∗} ⟩ = ∫_{S} ⟨ k (t, s) f (s) , y^{∗} ⟩ dν (s) = ⟨ y^{∗}, (Kf) (t) ⟩ .

Hence (3.7) holds. Thus, by Theorem 3.6, K maps E (S, X) into L_{∞} (T, Y).

Fix f ∈ L_{∞} (S, X). To see that Kf is σ (Z^{∗}, Z)-measurable, fix a sequence {f_{n}}_{n∈ℕ} from E (S, X) that converges a.e. to f and ∥f_{n}∥_{L_{∞}(S,X)} ≤ ∥f∥_{L_{∞}(S,X)} for each n ∈ ℕ. Then, by the Lebesgue Dominated Convergence Theorem, for each y^{∗} ∈ Z and for a.e. t ∈ T,

⟨ y^{∗}, (Kf) (t) ⟩ = ∫_{S} ⟨ f (s) , k^{∗} (t, s) y^{∗} ⟩ dν (s)

= lim_{n→∞} ∫_{S} ⟨ f_{n} (s) , k^{∗} (t, s) y^{∗} ⟩ dν (s) = lim_{n→∞} ⟨ y^{∗}, (Kf_{n}) (t) ⟩

and the latter functions ⟨ y^{∗}, (Kf_{n}) (·) ⟩ are μ-measurable functions by condition (C_{0}). Furthermore, by (3.11)

sup_{t∈T_{0}} ∥(Kf) (t)∥_{Z^{∗}} = sup_{t∈T_{0}} sup_{y^{∗}∈B(Z)} |∫_{S} ⟨ k (t, s) f (s) , y^{∗} ⟩ dν (s)| ≤ C_{∞} ∥f∥_{L_{∞}(S,X)} ;

thus, Kf ∈ L_{∞}^{w^{∗}} (T, Z^{∗}) and the K of (3.8) is of norm at most C_{∞} \underline{Proof of (i)} Assume that c_{0} does not isomorphically embed into Y.

Fix f ∈ L_{∞} (S, X). By Lemma 2.3, there is a sequence {f_{n}}_{n=1}^{∞} from E (S, X) so that

f (s) = ∑_{n=1}^{∞} f_{n} (s)

∑_{n=1}^{∞} ∥f_{n} (s)∥_{X} ≤ 2 ∥f∥_{L_{∞}(S,X)}

for a.e. s ∈ S. Since each f_{n} is in E (S, X), by (3.7) and condition (C_{0}), there is T_{1} ∈ Σ_{T}^{full}, with T_{1} ⊆ T_{0}, so that for each t ∈ T_{1} and each n ∈ ℕ

\begin{aligned}(Kf_{n}) (t) & = ∫_{S} k (t, s) f_{n} (s) dν (s) ∈ Y \\ Kf_{n} & ∈ L_{∞} (T, Y) . \end{aligned}  (3.12)

Fix t ∈ T_{1}. By the Lebesgue Dominated Convergence Theorem,

(Kf) (t) = lim_{m→∞} ∑_{n=1}^{m} (Kf_{n}) (t) in the σ (Z^{∗}, Z) -topology  (3.13)

since, for each y^{∗} ∈ Z,

⟨ y^{∗}, (Kf) (t) ⟩ = ∫_{S} ⟨ f (s) , k^{∗} (t, s) y^{∗} ⟩ dν (s)

= lim_{m→∞} ∫_{S} ⟨ ∑_{n=1}^{m} f_{n} (s) , k^{∗} (t, s) y^{∗} ⟩ dν (s) = lim_{m→∞} ⟨ ∑_{n=1}^{m} (Kf_{n}) (t) , y^{∗} ⟩ .

It suffices to show that the series in (3.13) converges also in the Z^{∗}-norm topology; thus, since Y does not contain c_{0}, it suffices to show that

∑_{n=1}^{∞} |⟨ (Kf_{n}) (t) , y^{∗} ⟩| < ∞ for each y^{∗} ∈ Y^{∗}  (3.14)

by a theorem of Bessaga and Pełczyński (cf., e.g., [4, Thm. V.8]).

For each y^{∗} ∈ Z,

∑_{n=1}^{∞} |⟨ (Kf_{n}) (t) , y^{∗} ⟩| = ∑_{n=1}^{∞} |∫_{S} ⟨ k (t, s) f_{n} (s) , y^{∗} ⟩ dν (s)|

≤ ∑_{n=1}^{∞} ∫_{S} |⟨ f_{n} (s) , k^{∗} (t, s) y^{∗} ⟩| dν (s)

≤ ∑_{n=1}^{∞} ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∥f_{n} (s)∥_{X} dν (s)

≤ ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} (∑_{n=1}^{∞} ∥f_{n} (s)∥_{X}) dν (s)

≤ 2 ∥f∥_{L_{∞}(S,X)} C_{∞} ∥y^{∗}∥_{Y^{∗}} .

Thus the mapping

Z ∋ y^{∗} ⟶^{U} {⟨ (Kf_{n}) (t) , y^{∗} ⟩}_{n=1}^{∞} ∈ ℓ_{1}

is a bounded linear operator. Fix {α_{n}}_{n=1}^{∞} ∈ B (ℓ_{∞}) and m ∈ ℕ. If y^{∗} ∈ B (Z) then

|⟨ ∑_{n=1}^{m} α_{n} (Kf_{n}) (t) , y^{∗} ⟩| = |∑_{n=1}^{m} α_{n} ⟨ (Kf_{n}) (t) , y^{∗} ⟩| ≤ ∥U∥_{B(Z,ℓ_{1})}

and so

∥∑_{n=1}^{m} α_{n} (Kf_{n}) (t)∥_{Y} ≤ τ ∥U∥_{B(Z,ℓ_{1})} .

Thus (3.14) holds (cf., e.g., [4, Thm. V.6]). \underline{Proof of (ii)} Assume that condition (ii) holds.

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 13

Fix f ∈ L_{∞} (S, X). Choose a sequence {f_{n}}_{n=1}^{∞} from E (S, X) such that

lim_{n→∞} f_{n} (s) = f (s) for a.e. s ∈ S

∥f_{n}∥_{L_{∞}(S,X)} ≤ ∥f∥_{L_{∞}(S,X)} for each n ∈ ℕ .

As in the proof of (i), since the f_{n}’s are in E (S, X), there is T_{1} ∈ Σ_{T}^{full}, with T_{1} ⊆ T_{0}, so that (3.12) holds for each t ∈ T_{1} and each n ∈ ℕ. It suffices to show that {(Kf_{n}) (t)}_{n=1}^{∞} converges in Z^{∗}-norm to (Kf) (t) for each t ∈ T_{1}.

Fix t ∈ T_{1}. Fix δ > 0 and let B_{n} := {s ∈ S: ∥f (s) − f_{n} (s)∥_{X} > δ} and A_{n} := ∪_{k=n}^{∞}B_{k}. Then

∥(Kf) (t) − (Kf_{n}) (t)∥_{Z^{∗}} = sup_{y^{∗}∈B(Z)} |∫_{S} ⟨ k (t, s) (f (s) − f_{n} (s)) , y^{∗} ⟩ dν (s)|

≤ sup_{y^{∗}∈B(Z)} ∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∥f (s) − f_{n} (s)∥_{X} dν (s)

≤ sup_{y^{∗}∈B(Z)} [δ ∫_{B_{n}^{C}} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s) + 2 ∥f∥_{L_{∞}(S,X)} ∫_{B_{n}} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s)]

≤ δ C_{∞} + 2 ∥f∥_{L_{∞}(S,X)} [ sup_{y^{∗}∈B(Z)} ∫_{A_{n}} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} dν (s)] .

Note that ∩_{n=1}^{∞}A_{n} ⊆ {s ∈ S: f_{n} (s) does not converge to f (s)}. Thus, by the equi-integrability assumption, {(Kf_{n}) (t)}_{n=1}^{∞} converges in Z^{∗}-norm to (Kf) (t). □

The following example illustrates the limitations on the conclusions in Theorems 3.6 and 3.11.

Example 3.13. Let X = ℂ and Y = c_{0}. Thus

B (X, Y) ≃ c_{0} and B (Y^{∗}, X^{∗}) ≃ ℓ_{∞} .

Let S = T = ℝ. Define

k_{0} : ℝ → c_{0}  k_{0} (·) := ∑_{n=1}^{∞} e_{n}1_{I_{n}} (·)
k: T × S → B (X, Y)  k (t, s) := k_{0} (t − s)

where {e_{n}}_{n=1}^{∞} is the standard unit vector basis of c_{0} and I_{n} = [n − 1, n).

Since k_{0} ∈ L_{∞} (ℝ, c_{0}), for each f ∈ L_{1} (S, X) the Bochner integral

(Kf) (t) := ∫_{S} k (t, s) f (s) ds = ∫_{ℝ} k_{0} (t − s) f (s) ds = (k_{0} ∗ f) (t)

exists for each t ∈ T; furthermore, Kf ∈ L_{∞} (T, Y) and Kf is uniformly continuous. From this it follows that k satisfies condition (C_{0}). The kernel k also satisfies condition (C_{∞}) with Z = Y^{∗} and T_{0} = T since for each y^{∗} ∈ Y^{∗} ≃ ℓ_{1} and t ∈ T

k^{∗} (t, s) y^{∗} = ∑_{n=1}^{∞} y^{∗} (e_{n}) 1_{I_{n}} (t − s) for each s ∈ S

∫_{S} ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ds = ∑_{n=1}^{∞} |y^{∗} (e_{n})| ≤ 1 ∥y^{∗}∥_{Y^{∗}} .

14 GIRARDI AND WEIS

Theorem 3.6 gives that K (L_{∞}^{0} (S, X)) ⊆ L_{∞} (T, Y). However, K (E (S, X)) ⊈ L_{∞}^{0} (T, Y). Indeed, (K1_{I_{1}}) (n) = e_{n} for each n ∈ ℕ and so, since K1_{I_{1}} is uniformly continuous, K1_{I_{1}} does not satisfy (2) of Proposition 2.2 and so K1_{I_{1}} ∉ L_{∞}^{0} (T, Y).

Theorem 3.11 gives that K (L_{∞} (S, X)) ⊆ L_{∞}^{w^{∗}} (T, Y^{∗∗}). However, K (L_{∞} (S, X)) ⊈ L_{∞} (T, Y). Indeed, consider f = 1_{(−∞,0)} ∈ L_{∞} (S, X). If n ∈ ℕ and 0 < δ ≤ 1 then

(Kf) (n − δ) = δe_{n} + ∑_{j∈ℕ}e_{n+j} ∈ Y^{∗∗} \ Y .

Thus Kf ∉ L_{∞} (T, Y). □

4. REMARKS ON DUALITY AND WEAK CONTINUITY

The remarks in this section explore the duality and weak continuity of K. For this, dual versions of the four conditions in Section 3 are needed.

Definition 4.1. Let (C) (possibly with respect to a subspace Z of Y^{∗}) be one of the four conditions in Section 3 on a kernel

k: T × S → B (X, Y) .

Then k satisfies condition (C^{∗}) (possibly with respect to a subspace Z of X^{∗∗}) provided the mapping

k̃: S × T → B (Y^{∗}, X^{∗})
k̃ (s, t) := [k (t, s)]^{∗}

satisfies condition (C) (possibly with respect to a subspace Z of X^{∗∗}).

For example, k: T × S → B (X, Y) satisfies condition (C_{0}^{∗}) provided that for each B ∈ Σ_{T}^{finite} and each y^{∗} ∈ Y^{∗}

• there is S_{B,y^{∗}} ∈ Σ_{S}^{full} so that if s ∈ S_{B,y^{∗}} then the Bochner integral

∫_{B} k^{∗} (t, s) y^{∗} dμ (t)

exists

• the mapping

S_{B,y^{∗}} ∋ s → ∫_{B} k^{∗} (t, s) y^{∗} dμ (t) ∈ X^{∗}

defines a measurable function from S into X^{∗}. □

Remark 4.2. Let 1 < p < ∞ and \frac{1}{p} + \frac{1}{p′} = 1. Let k: T × S → B (X, Y) be so that (i) k satisfies conditions (C_{0}), (C_{1}), (C_{∞}^{0}) with respect to Z = Y^{∗}, and (C_{0}^{∗}) (ii) for each y^{∗} ∈ Y^{∗} and each x ∈ X, the mapping T × S ∋ (t, s) → ⟨ k (t, s) x, y^{∗} ⟩ ∈ 𝕂 is

product measurable (iii) for each y^{∗} ∈ Y^{∗}, the mapping T × S ∋ (t, s) → ∥k^{∗} (t, s) y^{∗}∥_{X^{∗}} ∈ ℝ is product measurable.

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 15

By Theorem 3.8 (with Z = Y^{∗}), there is a bounded linear operator

K: L_{p} (S, X) → L_{p} (T, Y)  (4.1)

defined by

(Kf) (·) = ∫_{S} k (·, s) f (s) dν (s) ∈ L_{p} (T, Y) for f ∈ E (S, X) .

Note that k satisfies (iv) (C_{1}^{∗}) by (iii) and the fact that k satisfies (C_{∞}^{0}) with respect to Z = Y^{∗}

(v) (C_{∞}^{0∗}) with respect to Z = X since k satisfies (C_{1}). So by Theorem 3.8 (with Z = X), there is a bounded linear operator

K̃: L_{p′} (T, Y^{∗}) → L_{p′} (S, X^{∗})

defined by

(K̃g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{p′} (S, X^{∗}) for g ∈ E (T, Y^{∗}) .

Note that

K^{∗}g = K̃g for each g ∈ L_{p′} (T, Y^{∗})

since if g = y^{∗}1_{B} ∈ E (T, Y^{∗}) ⊂ L_{p′} (T, Y^{∗}) and f = x1_{A} ∈ E (S, X) ⊂ L_{p} (S, X)

\begin{aligned}⟨ f , K^{∗}g ⟩ & = ⟨ Kf , g ⟩ \\ & = ∫_{T} ⟨ ∫_{S} k (t, s) x1_{A} (s) dν (s) , y^{∗}1_{B} (t) ⟩ dμ (t) \\ & = ∫_{T} ∫_{S} ⟨ k (t, s) x1_{A} (s) , y^{∗}1_{B} (t) ⟩ dν (s) dμ (t) \\ & = ∫_{S} ∫_{T} ⟨ x1_{A} (s) , k^{∗} (t, s) y^{∗}1_{B} (t) ⟩ dμ (t) dν (s) \\ & = ∫_{S} ⟨ x1_{A} (s) , ∫_{T} k^{∗} (t, s) y^{∗}1_{B} (t) dμ (t) ⟩ dν (s) \\ & = ⟨ f , K̃g ⟩ \end{aligned}  (4.2)

where assumption (ii) helps justify the use of Fubini’s theorem. Thus (K^{∗}g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{p′} (S, X^{∗}) for g ∈ E (T, Y^{∗})

and K^{∗} maps L_{p′} (T, Y^{∗}) into L_{p′} (S, X^{∗}). Thus the K in (4.1) is

σ (L_{p} (S, X) , L_{p′} (S, X^{∗})) –to–σ (L_{p} (T, Y) , L_{p′} (T, Y^{∗}))

continuous. □

Remark 4.3. Let k: T × S → B (X, Y) be so that • k satisfies conditions (C_{0}), (C_{1}), and (C_{0}^{∗})

• condition (ii) of Remark 4.2 holds.

16 GIRARDI AND WEIS

By Theorem 3.4, there is a bounded linear operator

K: L_{1} (S, X) → L_{1} (T, Y)  (4.3)

defined by

(Kf) (·) = ∫_{S} k (·, s) f (s) dν (s) ∈ L_{1} (T, Y) for f ∈ E (S, X) .

Since k satisfies condition (C_{1}), it satisfies condition (C_{∞}^{0∗}) with respect to Z = X; thus, by Theorem 3.6, there is a bounded linear operator

K̃: L_{∞}^{0} (T, Y^{∗}) → L_{∞} (S, X^{∗})

defined by

(K̃g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{∞} (S, X^{∗}) for g ∈ E (T, Y^{∗}) .

Note that

K^{∗}g = K̃g for each g ∈ L_{∞}^{0} (T, Y^{∗})

since if g = y^{∗}1_{B} ∈ E (T, Y^{∗}) ⊂ L_{∞}^{0} (T, Y^{∗}) and f = x1_{A} ∈ E (S, X) ⊂ L_{1} (S, X) then the calculation in (4.2) shows that ⟨ f, K^{∗}g ⟩ = ⟨ f, K̃g ⟩. Thus

(K^{∗}g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{∞} (S, X^{∗}) for g ∈ E (T, Y^{∗})

and K^{∗} maps L_{∞}^{0} (T, Y^{∗}) into L_{∞} (S, X^{∗}). Thus the K in (4.3) is

σ (L_{1} (S, X) , L_{∞} (S, X^{∗})) –to–σ (L_{1} (T, Y) , L_{∞}^{0} (T, Y^{∗}))

continuous. □

Remark 4.4. Let k: T × S → B (X, Y) be so that • k satisfies conditions (C_{0}), (C_{1}), (C_{0}^{∗}), and (C_{∞}^{∗}) with respect to Z = X • condition (ii) of Remark 4.2 holds

• X^{∗} does not contain c_{0}. By Theorem 3.4, there is a bounded linear operator

K: L_{1} (S, X) → L_{1} (T, Y)  (4.4)

defined by

(Kf) (·) = ∫_{S} k (·, s) f (s) dν (s) ∈ L_{1} (T, Y) for f ∈ E (S, X) .

By Theorem 3.11, there is a bounded linear operator

K̃: L_{∞} (T, Y^{∗}) → L_{∞} (S, X^{∗})

defined by, for some S_{0} ∈ Σ_{S}^{full},

⟨ x, (K̃g) (s) ⟩ = ∫_{T} ⟨ x, k^{∗} (t, s) g (t) ⟩ dμ (t)
for g ∈ L_{∞} (T, Y^{∗}) and s ∈ S_{0} and x ∈ X .

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 17

Note that

K^{∗}g = K̃g for each g ∈ L_{∞} (T, Y^{∗})

since if g ∈ L_{∞} (T, Y^{∗}) and f = x1_{A} ∈ E (S, X) ⊂ L_{1} (S, X) then

⟨ f , K^{∗}g ⟩ = ⟨ Kf , g ⟩
= ∫_{T} ⟨ ∫_{S} k (t, s) x1_{A} (s) dν (s) , g (t) ⟩ dμ (t)
= ∫_{T} ∫_{S} ⟨ k (t, s) x , g (t) ⟩ 1_{A} (s) dν (s) dμ (t)
= ∫_{S} ∫_{T} ⟨ x , k^{∗} (t, s) g (t) ⟩ 1_{A} (s) dμ (t) dν (s)
= ∫_{S} ⟨ x , (K̃g) (s) ⟩ 1_{A} (s) dν (s)
= ⟨ f , K̃g ⟩

where assumption (ii) helps justify the use of Fubini’s theorem. Thus, by (3.7),

(K^{∗}g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{∞} (S, X^{∗}) for g ∈ E (T, Y^{∗})

and K^{∗} maps L_{∞} (T, Y^{∗}) into L_{∞} (S, X^{∗}). Thus the K in (4.4) is

σ (L_{1} (S, X) , L_{∞} (S, X^{∗})) –to–σ (L_{1} (T, Y) , L_{∞} (T, Y^{∗}))

continuous. □

Remark 4.5. Let k: T × S → B (X, Y) be so that • k satisfies conditions (C_{0}), (C_{∞}) with respect to Z = Y^{∗}, and (C_{0}^{∗})

• conditions (ii) and (iii) of Remark 4.2 hold. Then (3.9) of Theorem 3.11, with Z = Y^{∗}, defines a bounded linear operator

K: L_{∞} (S, X) → L_{∞}^{w^{∗}} (T, Y^{∗∗}) .  (4.5)

Note that k satisfies condition (C_{1}^{∗}) by (iii) and the fact that k satisfies condition (C_{∞}) with respect to Z = Y^{∗}. So by Theorem 3.4

(K̃g) (·) := ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{1} (S, X^{∗}) for g ∈ E (T, Y^{∗})

defines a bounded linear operator

K̃: L_{1} (T, Y^{∗}) → L_{1} (S, X^{∗}) .

Note that

K̃g = K^{∗}g for each g ∈ L_{1} (T, Y^{∗})

since for g = y^{∗}1_{B} ∈ E (T, Y^{∗}) ⊂ L_{1} (T, Y^{∗}) and f ∈ L_{∞} (S, X)

⟨ f , K^{∗}g ⟩ = ⟨ Kf , g ⟩
= ∫_{T} ⟨ y^{∗} , (Kf) (t) ⟩ 1_{B} (t) dμ (t)

= ∫_{T} [∫_{S} ⟨ k (t, s) f (s) , y^{∗} ⟩ dν (s)] 1_{B} (t) dμ (t)
= ∫_{S} ∫_{T} ⟨ f (s) , k^{∗} (t, s) y^{∗}1_{B} (t) ⟩ dμ (t) dν (s)
= ∫_{S} ⟨ f (s) , ∫_{T} k^{∗} (t, s) y^{∗}1_{B} (t) dμ (t) ⟩ dν (s)
= ∫_{S} ⟨ f (s) , (K̃g) (s) ⟩ dν (s)
= ⟨ f , K̃g ⟩

where assumption (ii) helps justify the use of Fubini’s theorem. Thus

(K^{∗}g) (·) = ∫_{T} k^{∗} (t, ·) g (t) dμ (t) ∈ L_{1} (S, X^{∗}) for each g ∈ E (T, Y^{∗})

and K^{∗} maps L_{1} (T, Y^{∗}) into L_{1} (S, X^{∗}). So the K of (4.5) is σ (L_{∞} (S, X) , L_{1} (S, X^{∗})) –to–σ (L_{∞}^{w^{∗}} (T, Y^{∗∗}) , L_{1} (T, Y^{∗}))

continuous. Thus, since E (S, X) (resp. the Schwartz class S (ℝ^{N}, X) in the case S = ℝ^{N}) is σ (L_{∞} (S, X) , L_{1} (S, X^{∗}))-dense in L_{∞} (S, X), many of the properties of K are determined by its restriction to E (S, X) (resp. S (ℝ^{N}, X)).

If furthermore Y does not contain c_{0}, then (3.9) of Theorem 3.11, with Z = Y^{∗}, defines a bounded linear operator

K: L_{∞} (S, X) → L_{∞} (T, Y)

that is σ (L_{∞} (S, X) , L_{1} (S, X^{∗})) –to–σ (L_{∞} (T, Y) , L_{1} (T, Y^{∗})) continuous. □

5. CONVOLUTION OPERATORS

The results thus far are now applied to convolution operators on T = S = R^{N} (endowed with the Lebesgue measure).

Corollary 5.1. Let Z be a subspace of Y^{∗} that τ-norms Y. Let k: ℝ^{N} → B (X, Y) be strongly measurable on X and k^{∗}: ℝ^{N} → B (Y^{∗}, X^{∗}) be strongly measurable on Z and

∫_{ℝ^{N}} ∥k (s) x∥_{Y} ds ≤ C_{1} ∥x∥_{X} < ∞  for each x ∈ X  (5.1)
∫_{ℝ^{N}} ∥k^{∗} (s) y^{∗}∥_{X^{∗}} ds ≤ C_{∞} ∥y^{∗}∥_{Y^{∗}} < ∞  for each y^{∗} ∈ Z .  (5.2)

Then the convolution operator

K: E (ℝ^{N}, X) → L_{0} (ℝ^{N}, Y )

defined by

(Kf) (t) := ∫_{R^{N}} k (t − s) f (s) ds for f ∈ E (ℝ^{N}, X)  (5.3)

extends to a bounded linear operator • K_{p}: L_{p} (ℝ^{N}, X) → L_{p} (ℝ^{N}, Y ) for 1 ≤ p < ∞,

• K_{∞}^{0} : L_{∞}^{0} (ℝ^{N}, X) → L_{∞}^{0} (ℝ^{N}, Y ),
• and, if Y does not contain c_{0}, then to K_{∞}: L_{∞} (ℝ^{N}, X) → L_{∞} (ℝ^{N}, Y ) satisfying

⟨ y^{∗}, (K_{∞}f) (t) ⟩ = ∫_{R^{N}} ⟨ k (t − s) f (s) , y^{∗} ⟩ dν (s)

for f ∈ L_{∞} (ℝ^{N}, X) and t ∈ ℝ^{N} and y^{∗} ∈ Z .

Furthermore,

∥K_{p}∥_{L_{p}→L_{p}} ≤ (C_{1})^{\frac{1}{p}} (τC_{∞})^{\frac{1}{p′}}

for 1 ≤ p ≤ ∞ and ∥K_{∞}^{0}∥_{L_{∞}^{0}→L_{∞}^{0}} ≤ τC_{∞}.

Remark 5.2. In Corollary 5.1, if Z = Y^{∗} and either • 1 < p < ∞

• p = 1 and X^{∗} does not contain c_{0}
• p = ∞ and Y does not contain c_{0},

then the dual operator

K_{p}^{∗}: [L_{p} (ℝ^{N}, Y )]^{∗} → [L_{p} (ℝ^{N}, X)]^{∗}

has the form

(K_{p}^{∗}g) (s) = ∫_{ℝ^{N}} k^{∗} (t − s) g (t) dt ∈ L_{p′} (ℝ^{N}, X^{∗}) for g ∈ E (ℝ^{N}, Y^{∗})

and K_{p}^{∗} maps L_{p′} (ℝ^{N}, Y^{∗}) into L_{p′} (ℝ^{N}, X^{∗}) and thus K_{p} is

σ (L_{p} (ℝ^{N}, X) , L_{p′} (ℝ^{N}, X^{∗})) –to–σ (L_{p} (ℝ^{N}, Y ) , L_{p′} (ℝ^{N}, Y^{∗}))

continuous.

Proof of Corollary 5.1 and Remark 5.2. If f = x1_{A} ∈ E (ℝ^{N}, X), then Kf = [k (·) x] ∗ 1_{A} with k (·) x ∈ L_{1} (ℝ^{N}, Y ) and 1_{A} ∈ L_{∞} (ℝ^{N}, ℝ). Thus for each f ∈ E (ℝ^{N}, X): the Bochner integral in (5.3) exists for each t in ℝ^{N}, Kf is a uniformly continuous function from ℝ^{N} to Y, and Kf vanishes at infinity. Thus K (E (ℝ^{N}, X)) ⊂ L_{∞}^{0} (ℝ^{N}, Y ).

It is straightforward to verify that the kernel

k_{0} : ℝ^{N} × ℝ^{N} → B (X, Y)
k_{0} (t, s) := k (t − s)

satisfies conditions: (C_{0}), (C_{1}), (C_{∞}) with respect to Z with T_{0} = ℝ^{N}, (C_{0}^{∗}), (C_{∞}^{∗}) with respect to Z = X, and (ii) and (iii) of Remark 4.2.

The corollary now follows from: Theorems 3.4, 3.6, 3.8, 3.11 and Remarks 4.2, 4.4, 4.5. □

Remark 5.3 (on Corollary 5.1).

(a) The proof shows that if k is strongly measurable on X and (5.1) holds, then one has the extension K_{1} : L_{1} (S, X) → L_{1} (T, Y). This case was shown in [6, Theorem V.1.14]. It is also pointed out there that, for 1 < p < ∞, condition (5.2) cannot be omitted.

(b) Under the stronger assumption that k ∈ L_{1} (ℝ^{N}, B (X, Y)), for 1 ≤ p ≤ ∞ the Bochner integrals

(K_{p}f) (t) := ∫_{ℝ^{N}} k (t − s) f (s) ds
f ∈ L_{p} (ℝ^{N}, X) ∩ L_{∞} (ℝ^{N}, X) and t ∈ ℝ^{N}

exist and define a bounded linear operator

K_{p}: L_{p} (ℝ^{N}, X) → L_{p} (ℝ^{N}, Y ) .

This fact is well-known and easy to show; indeed, for f ∈ L_{p} (ℝ^{N}, X) ∩ L_{∞} (ℝ^{N}, X) and f_{s} (t) := f (t − s),

∫_{ℝ^{N}} ∥k (t − s) f (s)∥_{Y} ds = ∫_{ℝ^{N}} ∥k (s) f_{s} (t)∥_{Y} ds ≤ ∥k∥_{L_{1}(ℝ^{N} ,B(X,Y))} ∥f∥_{L_{∞}(ℝ^{N} ,X)}

for each t ∈ ℝ^{N} and

∥(Kf) (·)∥_{L_{p}(ℝ^{N} ,Y)} ≤ ∫_{ℝ^{N}} ∥k (s) f_{s} (·)∥_{L_{p}(ℝ^{N} ,Y)} ds

≤ ∫_{ℝ^{N}} ∥k (s)∥_{B(X,Y)} ∥f_{s} (·)∥_{L_{p}(ℝ^{N} ,X)} ds
= ∥k∥_{L_{1}(ℝ^{N} ,B(X,Y))} ∥f∥_{L_{p}(ℝ^{N} ,X)} ;

thus, ∥K_{p}∥_{L_{p}→L_{p}} ≤ ∥k∥_{L_{1}(ℝ^{N} ,B(X,Y))}.

If, in addition, k satisfies (5.1) and k^{∗} satisfies (5.2) with Z = Y^{∗}, then it was shown in [8, Lemma 4.5] that ∥K_{p}∥_{L_{p}→L_{p}} ≤ (C_{1})^{\frac{1}{p}} (C_{∞})^{\frac{1}{p′}}. □

References

[1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.

[2] Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995.

[3] J. Diestel and J. J. Uhl, Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977, With a foreword by B. J. Pettis, Mathematical Surveys, No. 15.

[4] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984.

[5] Gerald B. Folland, Real analysis, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication.

[6] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Publishing Co., Amsterdam, 1985, Notas de Matemática [Mathematical Notes], 104.

[7] Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on L_{p}(X) and geometry of Banach spaces, Journal of Functional Analysis 204 no. 2 (2003), 320–354.

[8] Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr. 251 (2003), 34–51.

[9] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin, 1977, Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.

[10] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

[11] Jan Prüss, Evolutionary integral equations and applications, Birkhäuser Verlag, Basel, 1993.

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

E-mail address: girardi@math.sc.edu

INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS 21

Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany

E-mail address: Lutz.Weis@math.uni-karlsruhe.de