Continuous linear operators on Orlicz-Bochner spaces

Marian Nowak

doi:10.1515/math-2019-0089

Article Open Access

Continuous linear operators on Orlicz-Bochner spaces

Marian Nowak

Published/Copyright: October 13, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let L^φ(X) denote the corresponding Orlicz-Bochner space and Tφ∧ denote the finest Lebesgue topology on L^φ(X). We examine different classes of ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operators T : L^φ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

Keywords: Orlicz-Bochner spaces; Lebesgue topologies; weakly compact operators; compact operators; weakly completely continuous operators; completely continuous operators

MSC 2010: 47B38; 46E40; 28A25

1 Introduction and preliminaries

Throughout the paper, (X, ∥ ⋅ ∥_X) and (Y, ∥ ⋅ ∥_Y) denote real Banach spaces and X^* and Y^* denote their Banach duals, respectively. By B_X we denote the closed unit ball in X. Let 𝓛(X, Y) stand for the Banach space of all bounded linear operators from X to Y, equipped with the uniform operator norm ∥ ⋅ ∥.

Continuous linear operators on Banach spaces of vector-valued function spaces (in particular, Orlicz-Bochner spaces L^φ(X) and Lebesgue-Bochner spaces L^p(X) (1 ≤ p ≤ ∞)) has been the object of much study (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]). Andrews ([5, Theorems 2 and 5], [6, Theorem 3]) proved the Dunford-Pettis-Phillips type theorems for compact and weakly compact operators from L¹(X) to a Banach space Y.

Now we recall the basic concepts and properties of Orlicz-Bochner spaces (see [11, 12, 14, 15, 16] for more details).

By a Young function we mean here a continuous convex mapping φ : [0, ∞) → [0, ∞) that vanishes only at 0 and φ(t)/t → 0 as t → 0 and φ(t)/t → ∞ as t → ∞. Let φ^* stand for the complementary Young function of φ in the sense of Young.

We assume that (Ω, Σ, μ) is a complete σ-finite measure space. Denote by Σ_f(μ) the δ-ring of sets A ∈ Σ with μ(A) < ∞. By L⁰(X) we denote the linear space of μ-equivalence classes of all strongly Σ-measurable functions f:Ω → X.

Let L^φ(X) (resp., L^φ) denote the Orlicz-Bochner space (resp., Orlicz space) defined by a Young function φ, i.e.,

Lφ(X):={f∈L0(X):∫Ωφ(λ∥f(ω)∥X)dμ<∞for someλ>0}=f∈L0(X):∥f(⋅)∥X∈Lφ.

Then L^φ(X), equipped with the topology 𝓣_φ of the norm

∥f∥φ:=inf{λ>0:∫Ωφ∥f(ω)∥Xλdμ≤1}

is a Banach space.

For a bounded linear operator T : L^φ(X) → Y let

m(A)(x):=T(1A⊗x)forA∈Σf(μ),x∈X.

One can easily show that m(A) ∈ 𝓛(X, Y) for A ∈ Σ_f(μ). Then the mapping m : Σ_f(μ) → 𝓛(X, Y) will be called the representing measure of T.

Recall that a subset H of L^φ(X) is said to be solid whenever ∥f₁(ω)∥_X ≤ ∥f₂(ω)∥_X μ-a.e. and f₁ ∈ L^φ(X), f₂ ∈ H imply f₁ ∈ H. A linear topology ξ on L^φ(X) is said to be locally solid if it has a local basis at 0 consisting of solid sets (see [16]).

Following [17, Definition 2.2], [13] we have

Definition 1.1

A locally solid topology ξ on L^φ(X) is said to be a Lebesgue topology if for a net (f_α) in L^φ(X), ∥f_α(⋅)∥_X ⟶(o) 0 in L^φ implies f_α → 0 in ξ.

In view of the super Dedekind completeness of L^φ one can restrict in the above definition to usual sequences (f_n) in L^φ(X) (see [17, Definition 2.2, p. 173]).

Note that for a sequence (f_n) in L^φ(X), ∥f_n(⋅)∥_X ⟶(o) 0 in L^φ if and only if ∥f_n(ω)∥_X → 0 μ-a.e. and ∥f_n(ω)∥_X ≤ u(ω) μ-a.e. for some 0 ≤ u ∈ L^φ.

For ε > 0 let U_φ(ε) = {f ∈ L^φ(X): ∫_Ω φ(∥f(ω)∥_X) dμ ≤ ε}. Then the family of all sets of the form:

⋃n=1∞∑i=1nUφ(εi), (*)

where (ε_n) is a sequence of positive numbers, is a local basis at 0 for a linear topology Tφ∧ on L^φ(X) (see [13, 16] for more details). Using [16, Lemma 1.1] one can show that the sets of the form (*) are convex and solid, so Tφ∧ is a locally convex-solid topology.

We now recall terminology and basic facts concerning the spaces of weak^*-measurable functions g : Ω → X^* (see [18, 19]). Given a function g : Ω → X^* and x ∈ X, let g_x(ω) = g(ω)(X) for ω ∈ Ω. By L⁰(X^*, X) we denote the linear space of the weak^*-equivalence classes of all weak^*-measurable functions g : Ω → X^*. In view of the super Dedekind completeness of L⁰ the set {|g_x| : x ∈ B_X} is order bounded in L⁰ for each g ∈ L⁰(X^*, X). Thus one can define the so called abstract norm ϑ : L⁰(X^*, X) → L⁰ by

ϑ(g):=sup{|gx|:x∈BX} inL0.

It is known that for f ∈ L⁰(X), g ∈ L⁰(X^*, X), the function 〈f, g〉:Ω → ℝ defined by 〈f, g〉(ω) = 〈f(ω), g(ω)〉 is measurable and

|〈f(ω),g(ω)〉|≤∥f(ω)∥Xϑ(g)(ω)μ-a.e.

Moreover, ϑ(g) = ∥g(⋅)∥_X^* for g ∈ L⁰(X^*). Let

Lφ∗(X∗,X):={g∈L0(X∗,X):ϑ(g)∈Lφ∗}.

Clearly L^{φ^*}(X^*) ⊂ L^{φ^*}(X^*, X). If, in particular, X^* has the Radon-Nikodym property (i.e., X is an Asplund space see [20, p. 213]), then L^{φ^*}(X^*, X) = L^{φ^*}(X^*). Note that every reflexive Banach space X is an Asplund space.

Let (Lφ(X),Tφ∧)∗ denote the topological dual of (L^φ(X), Tφ∧ ).

Now we present basic properties of the topology Tφ∧ on L^φ(X).

Theorem 1.1

Let φ be a Young function. Then the following statements hold:

Tφ∧ ⊂ 𝓣_φ and Tφ∧ = 𝓣_φ if φ satisfies the Δ₂-condition, i.e., φ(2t) ≤ dφ(t) for some d > 1 and all t ≥ 0.
Tφ∧ is the finest Lebesgue topology on L^φ(X).
(Lφ(X),Tφ∧)∗ = {F_g : g ∈ L^{φ^*}(X^*, X)},

where Fg(f)=∫Ω〈f(ω),g(ω)〉dμforf∈Lφ(X).
If X is an Asplund space, then the space (L^φ(X), Tφ∧ ) is strongly Mackey; hence Tφ∧ coincides with the Mackey topology τ(L^φ(X), L^{φ^*}(X^*)).
If a subset H of L^φ(X) is Tφ∧ -bounded, then sup_f∈H ∥f∥_φ < ∞.

Proof

(i)–(ii) This follows from [16, Theorem 6.1 and Theorem 6.3].

(iii) In view of [13, Corollary 4.4 and Theorem 1.2], we get (Lφ(X),Tφ∧)∗=Lφ(X)n∼ , where Lφ(X)n∼ stands for the order continuous dual of L^φ(X) (see [13, 18] for more details). According to [18, Theorem 4.1] Lφ(X)n∼ = {F_g : g ∈ L^{φ^*} (X^*, X)}. Thus the proof is complete.

(iv) See [13, Theorem 4.5].

(v) Assume that a subset H of L^φ(X) is Tφ∧ -bounded. Then by (iv) H is σ(L^φ(X), L^{φ^*}(X^*, X))-bounded. Hence in view of [21, Proposition 1.3], the set {∥f(⋅)∥_X : f ∈ H} in L^φ is σ(L^φ, L^{φ^*})-bounded. Since L^{φ^*} is a norming subset of (L^φ)^* (see [22, p. 12]), by [22, Lemma 1, p. 20], we get sup_f∈H ∥f∥_φ = sup_f∈H∥ ∥f(⋅)∥_X ∥_φ < ∞. □

The following result establishes relationships between different classes of linear operators on L^φ(X).

Proposition 1.2

For a linear operator T : L^φ(X) → Y consider the following statements:

T is ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous.
T is ( Tφ∧ , ∥ ⋅ ∥_Y)-sequentially continuous.
∥T(f_n)∥_Y → 0 if ∥f_n(ω)∥_X → 0 μ-a.e. and ∥f_n(ω)∥_X ≤ u(ω) μ-a.e. for some 0 ≤ u ∈ L^φ and all n ∈ ℕ.
For every y^* ∈ Y^*, y^* ∘ T ∈ (Lφ(X),Tφ∧)∗ .
T is (σ(L^φ(X), L^{φ^*}(X^*, X)), σ(Y, Y^*))-continuous.
T is (τ(L^φ(X), L^{φ^*}(X^*, X)), ∥ ⋅ ∥_Y)-continuous.

Then the following implications hold:

(i)⇒(ii)⇒(iii)⇒(iv)⇒(v)⇒(vi).

If, in particular, X is an Asplund space, then (vi)⇒(i), that is, all the statements (i)–(vi) are equaivalent.

Proof

(i)⇒(ii)⇒(iii) Obvious because Tφ∧ is a Lebesgue topology.

(iii)⇒(iv) Assume that (iii) holds. Then for every y^* ∈ Y^*, y^* ∘ T ∈ Lφ(X)c∼ , where Lφ(X)c∼ stands for the σ-order continuous dual of L^φ(X) (see [17] for more details). In view of the super Dedekind completeness of L⁰ we have Lφ(X)c∼=Lφ(X)n∼ (see [17]). Since Lφ(X)n∼=(Lφ(X),Tφ∧)∗ , the proof is complete.

(iv)⇒(v) See [23, Theorem 9.26].

(v))⇒(vi) See [24, Theorem 8.6.1]. □

Assume that X is an Asplund space. Then (vi)⇒(i) holds because Tφ∧ = τ(L^φ(X), L^{φ^*}(X^*, X)) (see Theorem 1.1).

In this paper, using the results of [21], concerning conditional σ(L^φ(X), L^{φ^*}(X^*, X))-compactness and relative σ(L^φ(X), L^{φ^*}(X^*, X))-compactness in L^φ(X), we examine different classes of ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operators T : L^φ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. We establish relationships among these classes of operators.

2 Order-weakly compact and order-almost weakly compact operators on L^φ(X)

Dodds [25] studied the class of order-weakly compact operators on Banach lattices (see also [23, Section 18]). Following [25] one can define order-weakly compact and order-almost weakly compact operators on Orlicz-Bochner spaces L^φ(X) (see [12]).

For 0 ≤ u ∈ L^φ, let I_u = {f ∈ L^φ(X) : ∥f(ω)∥_X ≤ u(ω) μ-a.e.}.

Definition 2.1

A bounded linear operator T : L^φ(X) → Y is said to be order-weakly compact (resp. order-almost weakly compact) if for every 0 ≤ u ∈ L^φ, the set T(I_u) is a relatively weakly compact (resp., conditionally weakly compact) set in Y.

Recall that a Banach space X is called almost reflexive if every bounded set in X is conditionally σ(X, X^*)-compact. The fundamental ℓ¹-Rosenthal theorem says that a Banach space X is almost reflexive if and only if it contains no isomorphic copy of ℓ¹. Moreover, X contains no isomorphic copy of ℓ¹ if and only if X^* has the weak Radon-Nikodym property (see [26]).

Proposition 2.1

Assume that a Banach space X is almost reflexive (resp., X is reflexive). Then for every 0 ≤ u ∈ L^φ, the set I_u is conditionally σ(L^φ(X), L^{φ^*}(X^*, X))-compact (resp., relatively σ(L^φ(X), L^{φ^*}(X^*))-compact).

Proof

Let 0 ≤ u ∈ L^φ. Then I_u is a norm bounded subset of L^φ(X) and for every v ∈ L^{φ^*}, we have uv ∈ L¹ and

pIu(v):=supf∈Iu∫Ω∥f(ω)∥X|v(ω)|dμ≤∫Ω|u(ω)v(ω)|dμ.

To show that p_{I_u} is an order continuous seminorm on L^{φ^*}, assume that (v_n) is a sequence in L^{φ^*} such that v_n ⟶(o) 0 in L^{φ^*}, i.e., v_n(ω) → 0 μ-a.e. and |v_n(ω)| ≤ v(ω) μ-a.e. for some 0 ≤ v ∈ L^{φ^*} and all n ∈ ℕ. Since u v ∈ L¹, by the Lebesgue dominated convergence theorem p_{I_u}(v_n) → 0. In view of [21, Proposition 1.1] (resp. [21, Proposition 1.1] and [22, Lemma 11(a), p. 31]) the set I_u is conditionally σ(L^φ, L^{φ^*})-compact (resp., relatively σ(L^φ, L^{φ^*})-compact). □

Theorem 2.2

Assume that a Banach space X is reflexive. Then every ( Tφ∧ ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y is order-weakly compact.

Proof

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator and 0 ≤ u ∈ L^φ. Then by Proposition 2.1 I_u is a relatively σ(L^φ(X), L^{φ^*}(X^*))-compact set in L^φ(X). Since T is (σ(L^φ(X), L^{φ^*}(X^*)), σ(Y, Y^*))-continuous, T(I_u) is relatively σ(Y, Y^*)-compact in Y. □

Using Proposition 2.1 and arguing as in the proof of Theorem 2.2, we get:

Theorem 2.3

Assume that a Banach space X is almost reflexive. Then every ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y is order-almost weakly compact.

3 Weakly compact and almost weakly compact operators on L^φ(X)

We say that a Young function ψ increases more rapidly than another φ (in symbols, φ ≺ ψ) if for arbitrary c > 0 there exists d > 0 such that cφ(t) ≤ 1d ψ(dt) for all t ≥ 0. Recall that a Young function φ satisfies the ▽₂-condition, if φ(t) ≤ 12d φ(dt) for some d > 1 and all t ≥ 0. It is known that φ satisfies the ▽₂-condition if and only if φ^* satisfies the Δ₂-condition (see [27, Theorem 2.2.3, pp. 22-23]).

The following results will be useful (see [27, Theorem 5.3.3, p. 171]).

Proposition 3.1

Let φ and ψ be Young functions such that φ ≺ ψ. Then L^ψ ⊂ L^φ and every norm bounded set in L^ψ is relatively σ(L^φ, L^{φ^*})-compact in L^φ.

Theorem 3.2

Let φ be a Young function. Then for a subset H of L^φ the following statements are equivalent:

H is conditionally σ(L^φ, L^{φ^*})-compact.
H is relatively σ(L^φ, L^{φ^*})-sequentially compact.
H is relatively σ(L^φ, L^{φ^*})-compact.
There exists a Young function ψ with φ ≺ ψ such that H ⊂ L^ψ and sup {∥u∥_ψ : u ∈ H} ≤ 1.

Proof

(i)⇔(ii) See [21, Proposition 1.1].

(ii)⇔(iii) This follows from [22, Lemma 11(a), p. 31].

(iii)⇔(iv) This follows from [28, Theorem 1.2]. □

Remark

For a finite measure space (Ω, Σ, μ), the equivalence (iii)⇔(iv) in Theorem 3.2 was established by Ando (see [29, Theorem 2]).

If φ ≺ ψ, then L^ψ(X) ⊂ L^φ(X) and 𝓣_φ |_L^ψ(X) ⊂ 𝓣_ψ. Let

iψ:Lψ(X)→Lφ(X)

stand for the inclusion map and

BLψ(X)={f∈Lψ(X):∥f∥ψ≤1}.

Recall that a bounded linear operator T from a Banach space Z to Y is said to be weakly compact (resp. almost weakly compact) if T(B_Z) is a relatively weakly compact (resp. conditionally weakly compact) set in Y.

Theorem 3.3

Assume that a Banach space X is reflexive and T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. Then for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is weakly compact.

Proof

Let ψ be a Young function with φ ≺ ψ. Then by Proposition 3.1 the set {∥f(⋅)∥_X : f ∈ B_L^ψ(X)} in L^φ is relatively σ(L^φ, L^{φ^*})-compact, and hence by Theorem 3.2 it is relatively σ(L^φ, L^{φ^*})-sequentially compact. By [21, Corollary 3.4 and Theorem 3.2] B_L^ψ(X) is relatively σ(L^φ(X), L^{φ^*}(X^*))-compact. Since T is (σ(L^φ(X), L^{φ^*}(X^*)), σ(Y, Y^*))-continuous, T(B_L^ψ(X)) is relatively σ(Y, Y^*)-compact, and hence T ∘ i_ψ is weakly compact. □

Corollary 3.4

Assume that a Banach space X is reflexive and Young function φ satisfies the ▽₂-condition. Then every ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y is weakly compact.

Theorem 3.5

Assume that a Banach space X is almost reflexive and T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. Then for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is almost weakly compact.

Proof

Let ψ be a Young function with φ ≺ ψ. Then by Proposition 3.1 the set {∥f(⋅)∥_X : f ∈ B_L^ψ(X)} in L^φ is relatively σ(L^φ, L^{φ^*})-compact, and hence by Theorem 3.2 it is conditionally σ(L^φ, L^{φ^*})-compact. By [21, Corollary 2.5] B_L^ψ(X) is conditionally σ(L^φ(X), L^{φ^*}(X^*, X))-compact. Since T is (σ(L^φ(X), L^{φ^*}(X^*, X)), σ(Y, Y^*))-continuous, T(B_L^ψ(X)) is conditionally σ(Y, Y^*)-compact, i.e., T ∘ i_ψ is almost weakly compact. □

Corollary 3.6

Assume that a Banach space X is almost reflexive and a Young function φ satisfies the ▽₂-condition. Then every ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y is almost weakly compact.

4 Weakly completely continuous operators on L^φ(X)

Definition 4.1

Assume that (Z, ξ) is a locally convex Hausdorff space. A (ξ, ∥ ⋅ ∥_Y)-continuous linear operator T : Z → Y is said to be weakly completely continuous if T maps weakly-Cauchy sequences in Z onto weakly-convergent sequences in Y.

Recall that a weakly completely continuous operator between Banach spaces is usually called a Dieudonné operator.

Theorem 4.1

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator and m : Σ_f(μ) → 𝓛(X, Y) be its representing measure. If T weakly completely continuous, then for every A ∈ Σ_f(μ), m(A) is a Dieudonné operator.

Proof

Assume that (x_n) is a σ(X, X^*)-Cauchy sequence in X and A ∈ Σ_f(μ). We shall show that (𝟙_A ⊗ x_n) is a σ(L^φ(X), L^{φ^*}(X^*, X))-Cauchy sequence in L^φ(X). Indeed, let g ∈ L^{φ^*}(X^*, X) be given. Then g_{x_n}(ω) → v_g(ω) μ-a.e., where ϑ(g) ∈ L^{φ^*}. Since |g_{x_n}(ω)| ≤ ϑ(g)(ω)∥x_n∥_X ≤ aϑ(g)(ω) μ-a.e., where a = sup_n ∥x_n∥_X < ∞, we get v_g(ω) ≤ aϑ(g)(ω) μ-a.e. Hence v_g ∈ L^{φ^*} and 𝟙_A v_g ∈ L¹. Note that 𝟙_A(ω)g_{x_n}(ω) → 𝟙_A(ω)v_g(ω) μ-a.e. and |𝟙_A(ω)g_{x_n}(ω)| ≤ a 𝟙_A(ω)ϑ(g)(ω) μ-a.e. for all n ∈ ℕ. Then by the Lebesgue dominated convergence theorem,

∫Ω〈(1A⊗xn)(ω),g(ω)〉dμ=∫Ω1A(ω)gxn(ω)dμ→∫Ω1A(ω)vg(ω)dμ.

This means that (𝟙_A ⊗ x_n) is a σ(L^φ(X), L^{φ^*}(X^*, X))-Cauchy sequence in L^φ(X). Since m(A)(x_n) = T(𝟙_A ⊗ x_n) for n ∈ ℕ, we obtain that (m(A)(x_n)) is a σ(Y, Y^*)-convergent sequence in Y, as desired. □

Theorem 4.2

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. Assume that for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is weakly compact. Then T is weakly completely continuous.

Proof

Assume that (f_n) is a σ(L^φ(X), L^{φ^*}(X^*, X))-Cauchy sequence in L^φ(X). Then the set {f_n : n ∈ ℕ} is conditionally σ(L^φ(X), L^{φ^*}(X^*, X))-compact in L^φ(X), and it follows that {∥f_n(⋅)∥_X : n ∈ ℕ} is a conditionally σ(L^φ, L^{φ^*})-compact set in L^φ (see [21, Theorem 2.2]). Then in view of Theorem 3.2 there exists a Young function ψ with φ ≺ ψ such that sup_n ∥f_n∥_ψ ≤ 1. It follows that the set {T(f_n) : n ∈ ℕ} is relatively σ(Y, Y^*)-compact in Y. Then there exists a subsequence (f_{k_n}) of (f_n) such that T(f_{k_n}) → y_o in σ(Y, Y^*) for some y_o ∈ Y. On the other hand, since T is (σ(L^φ(X), L^{φ^*}(X^*, X)), σ(Y, Y^*))-continuous, (T(f_n)) is a σ(Y, Y^*)-Cauchy sequence in Y. It follows that T(f_n) → y_o in σ(Y, Y^*). □

As a consequence of Theorem 4.2 we have:

Corollary 4.3

Assume that T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous. If T is weakly compact operator, then T is weakly completely continuous.

Theorem 4.4

Assume that a Banach space X is almost reflexive and T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. If T is weakly completely continuous, then for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is weakly compact.

Proof

Let ψ be a Young function such that φ ≺ ψ. Then by Proposition 3.1 the set {∥f(⋅)∥_X : f ∈ B_L^ψ(X)} in L^φ is relatively σ(L^φ, L^{φ^*})-compact, and hence it is conditionally σ(L^φ, L^{φ^*})-compact (see Theorem 3.2). In view of [21, Corollary 2.3] B_L^ψ(X) is conditionally σ(L^φ(X), L^{φ^*}(X^*, X))-compact. To show that T(B_L^ψ(X)) is relatively σ(Y, Y^*)-compact, assume that (y_n) is a sequence in T(B_L^ψ(X)), i.e., y_n = T(f_n), where f_n ∈ B_L^ψ(X). Then there exists a σ(L^φ(X), L^{φ^*}(X^*, X))-Cauchy subsequence (f_{k_n}) of (f_n). Hence T(f_{k_n}) → y_o in σ(Y, Y^*) for some y_o ∈ Y, and this means T(B_L^ψ(X)) is relatively σ(Y, Y^*)-sequentially compact in Y. By the Eberlein-Šmulian theorem, T(B_L^ψ(X)) is relatively σ(Y, Y^*)-compact, as desired. □

Corollary 4.5

Assume that a Banach space X is almost reflexive and a Young function φ satisfies the ▽₂-condition. Then for a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y the following statements are equivalent:

T is weakly completely continuous.
T is weakly compact.

Proof

(i)⇒(ii) It follows from Theorem 4.3.

(ii)⇒(i) See Corollary 4.2.

Following [24, Section 9.4] we have: ◼

Definition 4.2

A locally convex Hausdorff space (Z, ξ) is said to have the Dieudonné property if for every Banach space Y, every weakly completely continuous operator T : Z → Y maps ξ-bounded sets in Z onto relatively weakly compact sets in Y.

Corollary 4.6

Assume that a Banach space X is almost reflexive and a Young function φ satisfies the ▽₂-condition. Then the space (L^φ(X), Tφ∧ ) has the Dieudonné property.

Proof

It follows from Corollary 4.5 because every Tφ∧ -bounded set in L^φ(X) is 𝓣_φ-bounded (see Theorem 1.1). □

Definition 4.3

Assume that (Z, ξ) is a locally convex Hausdorff space. A (ξ, ∥ ⋅ ∥_Y)-contnuous linear operator T : Z → Y is said to be unconditionally converging if the series ∑n=1∞ T(z_n) converges unconditionally in Y whenever ∑n=1∞ |z^*(z_n)| < ∞ for every z^* ∈ (Z, ξ)^*.

Proposition 4.7

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. If T is weakly completely continuous, then T is unconditionally converging.

Proof

Assume that (f_n) is a sequence in L^φ(X) such that ∑n=1∞ | ∫_Ω 〈f_n(ω), g(ω)〉 dμ| < ∞ for all g ∈ L^{φ^*}(X^*, X). For a subsequence (f_{k_n}) of (f_n), let S_n = ∑n=1∞ f_{k_i}. Then (S_n) is a σ(L^φ(X), L^{φ^*}(X^*, X))-Cauchy sequence in L^φ(X). It follows that the series ∑n=1∞ T(f_{k_n}) is σ(Y, Y^*)-convergent in Y and in view of the Orlicz-Pettis theorem (see [20, p. 22]), the series ∑n=1∞ T(f_n) is unconditionally convergent. This means that T is unconditionally converging. □

5 Completely continuous operators on L^φ(X)

Definition 5.1

Assume that (Z, ξ) is a locally convex Hausdorff space. A (ξ, ∥ ⋅ ∥_Y)- continuous linear operator T : Z → Y is said to be completely continuous if ∥T(z_n)∥_Y → 0 whenever (z_n) converges weakly to 0 in Z.

Recall that a completely continuous operator between Banach spaces is usually called a Dunford-Pettis operator.

Theorem 5.1

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator and m : Σ_f(μ) → 𝓛(X, Y) be its representing measure. If T completely continuous, then for every A ∈ Σ_f(μ), m(A) is a Dunford-Pettis operator.

Proof

Assume that x_n → 0 in σ(X, X^*) and A ∈ Σ_f(μ). We shall show that 𝟙_A ⊗ x_n → 0 in σ(L^φ(X), L^{φ^*}(X^*, X)). Indeed, let g ∈ L^{φ^*}(X^*, X) be given. Note that 𝟙_A(ω)g_{x_n}(ω) → 0 μ-a.e. and |𝟙_A(ω)g_{x_n}(ω)| ≤ 𝟙_A(ω)ϑ(g)(ω)∥x_n∥_X ≤ a 𝟙_A(ω)ϑ(g)(ω) μ-a.e. for all n ∈ ℕ, where a = sup_n ∥x_n∥_X < ∞. Since ϑ(g) ∈ L^{φ^*}, we get 𝟙_A ϑ(g) ∈ L¹. Hence by the Lebesgue dominated convergence theorem

∫Ω〈(1A⊗xn)(ω),g(ω)〉dμ=∫Ω1A(ω)gxn(ω)dμ→0.

It folows that ∥m(A)(x_n)∥_Y = ∥T(𝟙_A ⊗ x_n)∥_Y → 0. □

Bourgain [30, Proposition 1] showed that a bounded linear operator T : L¹ → Y (μ(Ω) < ∞) is Dunford-Pettis if and only if T restricted to L^p for some p ∈ (1, ∞] is compact. Now we extend this result to operator T : L^φ(X) → Y. We study the relationships between completely continuous operators T : L^φ(X) → Y and the compactness properties of T restricted to L^ψ(X), where φ ≺ ψ.

Theorem 5.2

Let T : L^φ(X) → Y be a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. Assume that for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is compact. Then T is completely continuous.

Proof

Assume that f_n → 0 in σ(L^φ(X), L^{φ^*}(X^*, X)). Then the set {f_n : n ∈ ℕ} is relatively σ(L^φ(X), L^{φ^*}(X^*, X))-sequentially compact in L^φ(X), and it follows that {∥f_n(⋅)∥_X : n ∈ ℕ} is relatively σ(L^φ, L^{φ^*})-sequentially compact set in L^φ (see [21, Theorem 3.3]). Then by Theorem 3.2 there exists a Young function ψ with φ ≺ ψ such that sup_n ∥f_n∥_ψ ≤ 1. It follows that {T(f_n) : n ∈ ℕ} is a relatively norm compact set in Y. Hence there exists a subsequence (f_{k_n}) of (f_n) and y_o ∈ Y such that ∥T(f_{k_n}) − y_o∥_Y → 0. On the other hand, since T is (σ(L^φ(X), L^{φ^*}(X^*, X)), σ(Y, Y^*))-continuous, we get T(f_n) → 0 in σ(Y, Y^*). Hence y_o = 0 and ∥T(f_{k_n})∥_Y → 0. This means that ∥T(f_n)∥_Y → 0. □

As a consequence of Theorem 5.2, we have:

Corollary 5.3

Assume that T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. If T is compact, then T is completely continuous.

Theorem 5.4

Assume that X is a reflexive Banach space and T : L^φ(X) → Y is a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator. If T is completely continuous, then for every Young function ψ with φ ≺ ψ, the operator T ∘ i_ψ : L^ψ(X) → Y is compact.

Proof

Let ψ be a Young function such that φ ≺ ψ. Then by Proposition 3.1 the set {∥f(⋅)∥_X : f ∈ B_L^ψ(X)} in L^φ is relatively σ(L^φ, L^{φ^*})-compact and hence it is relatively σ(L^φ, L^{φ^*})-sequentially compact (see Theorem 3.2). In view of [21, Corollary 3.4] B_L^ψ(X) is a relatively σ(L^φ(X), L^{φ^*}(X^*))-sequentially compact set in L^φ(X). To show that T(B_L^ψ(X)) is a relatively norm compact subset of Y, assume that (y_n) is a sequence in T(B_L^ψ(X)), i.e., y_n = T(f_n), where f_n ∈ B_L^ψ(X). Then there exists a subsequence (f_{k_n}) of (f_n) such that f_{k_n} → f_o in σ(L^φ(X), L^{φ^*}(X^*)) for some f_o ∈ L^φ(X). Hence ∥T(f_{k_n}) − T(f_o)∥_Y → 0 and this means that T(B_L^ψ(X)) is relatively compact in Y. □

Corollary 5.5

Assume that X is a reflexive Banach space and a Young function φ satisfies the ▽₂-condition. Then for a ( Tφ∧ , ∥ ⋅ ∥_Y)-continuous linear operator T : L^φ(X) → Y the following statements are equivalent:

T is completely continuous.
T is compact.

Proof

(i)⇒(ii) This follows from Theorem 5.4.

(ii)⇒(i) See Corollary 5.3. □

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Received: 2019-01-29

Accepted: 2019-08-15

Published Online: 2019-10-13

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0089

Keywords for this article

Orlicz-Bochner spaces; Lebesgue topologies; weakly compact operators; compact operators; weakly completely continuous operators; completely continuous operators

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Continuous linear operators on Orlicz-Bochner spaces

Article

Abstract

1 Introduction and preliminaries

Definition 1.1

Theorem 1.1

Proof

Proposition 1.2

Proof

2 Order-weakly compact and order-almost weakly compact operators on Lφ(X)

Definition 2.1

Proposition 2.1

Proof

Theorem 2.2

Proof

Theorem 2.3

3 Weakly compact and almost weakly compact operators on Lφ(X)

Proposition 3.1

Theorem 3.2

Proof

Remark

Theorem 3.3

Proof

Corollary 3.4

Theorem 3.5

Proof

Corollary 3.6

4 Weakly completely continuous operators on Lφ(X)

Definition 4.1

Theorem 4.1

Proof

Theorem 4.2

Proof

Corollary 4.3

Theorem 4.4

Proof

Corollary 4.5

Proof

Definition 4.2

Corollary 4.6

Proof

Definition 4.3

Proposition 4.7

Proof

5 Completely continuous operators on Lφ(X)

Definition 5.1

Theorem 5.1

Proof

Theorem 5.2

Proof

Corollary 5.3

Theorem 5.4

Proof

Corollary 5.5

Proof

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2 Order-weakly compact and order-almost weakly compact operators on L^φ(X)

3 Weakly compact and almost weakly compact operators on L^φ(X)

4 Weakly completely continuous operators on L^φ(X)

5 Completely continuous operators on L^φ(X)