# Bochner representable operators on Banach function spaces

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## Abstract

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of *E* and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in *E* onto relatively norm compact sets in *X*. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on *E*. Applications to Bochner representable operators on Orlicz spaces are given.

## Keywords

Banach function spaces Orlicz spaces Mixed topologies Bochner representable operators Compact operators## Mathematics Subject Classification

47B38 46B40 46E30 47B05## 1 Introduction and preliminaries

We assume that \((X,\Vert \cdot \Vert _X)\) is a real Banach space with the Banach dual \((X^*,\Vert \cdot \Vert _{X^*})\). Let \(B_X\) stand for the closed unit ball in *X*. For terminology concerning Riesz spaces and function spaces, we refer the reader to [1, 2, 9, 10, 20].

We assume that \((\Omega ,\Sigma ,\mu )\) is a complete \(\sigma \)-finite measure space. By \(\Sigma _f\) we denote the \(\delta \)-ring of all sets \(A\in \Sigma \) with \(\mu (A)<\infty \). Let \(L^0\) denote the corresponding space of equivalence classes of all \(\Sigma \)-measurable real functions on \(\Omega \). Then \(L^0\) is a super Dedekind complete Riesz space, equipped with the *F*-norm topology \(\mathcal{T}_0\) of convergence in measure on sets of finite measure.

*E*is an order ideal of \(L^0\) with \(\mathrm{supp}\,E=\Omega \) and \(\Vert \cdot \Vert _E\) is a Riesz norm. By \(\mathcal{T}_E\) we denote the topology of the norm \(\Vert \cdot \Vert _E\). For \(r>0\), let \(B_E(r):=\{u\in E:\Vert u\Vert _E\le r\}.\) The Köthe dual \(E'\) of

*E*is defined by:

*E*separates the points of

*E*and \(E^\sim _c\) can be identified with \(E'\) through the Riesz isomorphism \(E'\ni v\mapsto F_v\in E^\sim _c\), where

- (A)
\(E\subset L^1_{loc}\), i.e., \(A\in \Sigma _f\) implies \(u\mathbb {1}_A\in L^1\) for all \(u\in E\).

- (B)
*E*is perfect, i.e., \(E=E''\) (equivalently, \(\Vert \cdot \Vert _E\) satisfies both the \(\sigma \)-Fatou property and the \(\sigma \)-Levy property (see [1, Definition 3.14], [20, § 110 and Theorem 112.2]).

*A*) it follows that for every \(A\in \Sigma _f\), \(\mathbb {1}_A\in E'\). Moreover, from (

*B*) it follows that

*E*is \(\sigma (E,E')\)-sequentially complete (see [13, Proposition 1.2]). Note that if a subset

*H*of

*E*is \(\sigma (E,E')\)-bounded, then \(\sup _{u\in H}\Vert u\Vert _E<\infty \) (see [10, Lemma 1, p. 20], [9, Theorem 6.1.6]). By \((E')_a\) we denote the ideal in \(E'\) of all elements of order continuous norms, that is,

### Definition 1.1

*Bochner representable*, if there exists \(g\in E'(X)\) such that

## 2 Bochner representable operators on Banach spaces

It is known that every Bochner representable operator \(T:L^1\rightarrow X\) (where \(\mu (\Omega )<\infty )\) maps relatively \(\sigma (L^1,L^\infty )\)-compact sets onto relatively norm compact sets in *X* (see [5, Lemma 11, pp. 74–75]). Now we extend this result to Bochner representable operators \(T:E\rightarrow X\).

### Theorem 2.1

- (i)
*T*is \((\tau (E,E'),\Vert \cdot \Vert _X)\)-continuous. - (ii)
*T*maps relatively \(\sigma (E,E')\)-compact sets in*E*onto relatively norm compact sets in*X*.

### Proof

*T*is \((\sigma (E,E'),\sigma (X,X^*))\)-continuous (see [2, Theorem 9.26]), and hence

*T*is \((\tau (E,E'),\Vert \cdot \Vert _X)\)-continuous (see [8, Theorem 8.6.1]).

*H*is a relatively \(\sigma (E,E')\)-compact subset of

*E*. It follows that

*H*is relatively \(\sigma (E,E')\)-sequentially compact (see [10, Lemma 11, p. 31]). Since

*E*is \(\sigma (E,E')\)-sequentially complete, in view of [13, Proposition 1.1] the set \(\{u\Vert g(\cdot )\Vert _X:u\in H\}\) in \(L^1\) is uniformly integrable. Hence given \(\varepsilon >0\) there exist \(\Omega _0\in \Sigma _f\) and \(\delta >0\) such that

*X*-valued strongly \(\mu \)-measurable step functions on \(\Omega \) such that \(s_n(\omega )\rightarrow g(\omega )\) \(\mu \)-a.e. and \(\Vert s_n(\omega )\Vert _X\le \Vert g(\omega )\Vert _X\) \(\mu \)-a.e. for all \(n\in \mathbb {N}\) (see [7, Theorem 6, p. 4]). Hence \(s_n\in E'(X)\) for \(n\in \mathbb {N}\). By the Egorov theorem (see [7, Theorem 42, p. 18]), there exists \(A_0\in \Sigma \) with \(A_0\subset \Omega _0\), \(\mu (\Omega _0\smallsetminus A_0)\le \delta \) and \(\sup _{\omega \in A_0}\Vert s_n(\omega )-g(\omega )\Vert _X\rightarrow 0\). For every \(n\in \mathbb {N}\) define an operator \(T_n:E\rightarrow X\) by

*X*.

*T*(

*H*) is a relatively norm compact set in

*X*, as desired. \(\square \)

*natural mixed topology*on

*E*in the sense of Wiweger (see [3, 11, 12, 19], for more details). Then \(\mathcal{T}_0\big |_E\subset \gamma _E\subset \mathcal{T}_E\) and \(\gamma _E\) is the finest linear topology on

*E*that agrees with \(\mathcal{T}_0\) on every ball \(B_E(r)\), \(r>0\) (see [19, 2.2.2]). \(\mathcal{T}_E\) and \(\gamma _E\) have the same bounded sets in

*E*, and for a sequence \((u_n)\) in

*E*, \(u_n\rightarrow 0\) in \(\gamma _E\) if and only if \(u_n\rightarrow 0\) in \(\mathcal{T}_0\) and \(\sup _n\Vert u_n\Vert _E<\infty \) (see [19, Corollary, p. 56 and Theorem 2.6.1]). If, in particular, \(\mathrm{supp}\,(E')_a=\Omega \), then \(\gamma _E\) is a locally convex-solid Hausdorff topology and \(\gamma _E\) coincides with the mixed topology \(\gamma [\mathcal{T}_E,|\sigma |(E,(E')_a)]\) (see [11, Theorem 3.3]). Note that, then \((E,\gamma _E)\) is a generalized DF-space (see [15] for more details). In view of [11, Theorem 3.1], we have:

### Theorem 2.2

- (i)
*T*is \((\gamma _E,\Vert \cdot \Vert _X)\)-continuous and norm-compact operator. - (ii)
*T*is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, that is, there exists a \(\gamma _E\)-neighborhood*V*of 0 in*E*such that*T*(*V*) is a relatively norm compact set in*X*.

### Proof

*T*is \((\gamma _E,\Vert \cdot \Vert _X)\)-continuous. Choose a sequence \((s_n)\) of

*X*-valued strongly \(\mu \)-measurable step-functions on \(\Omega \) such that \(\Vert s_n(\omega )-g(\omega )\Vert _X\rightarrow 0\) \(\mu \)-a.e. and \(\Vert s_n(\omega )\Vert _X\le \Vert g(\omega )\Vert _X\) \(\mu \)-a.e. and all \(n\in \mathbb {N}\) (see [7, Theorem 6, p. 4]). Hence \(s_n\in (E')_a(X)\) and \(\Vert s_n(\omega )-g(\omega )\Vert _X\le 2\Vert g(\omega )\Vert _X\) \(\mu \)-a.e. for all \(n\in \mathbb {N}\). Let \(v_n(\omega )=\sup _{k\ge n}\Vert s_n(\omega )-g(\omega )\Vert _X\) for \(\omega \in \Omega \). Then \(2\Vert g(\omega )\Vert _X\ge v_n(\omega )\downarrow 0\) \(\mu \)-a.e., so \(\Vert \;\Vert s_n(\cdot )-g(\cdot )\Vert _X \;\Vert _{E'}\le \Vert v_n\Vert _{E'}\rightarrow 0\).

*T*is a compact operator.

(ii) Since \((E,\gamma _E)\) is a generalized DF-space, it is quasinormable (see [15, p. 422]). Hence in view of (i) by the Grothendieck’s classical results (see [15, p. 429]), we obtain that *T* is \((\gamma _E,\Vert \cdot \Vert )\)-compact. \(\square \)

### Remark 2.1

A related result to Theorem 2.2 can be found in [17, Corollary 4.8].

An important class of Banach function spaces are Orlicz spaces (see [10, 16] for more details). By a Young function we mean here s convex continuous mapping \(\Phi :[0,\infty )\rightarrow [0,\infty )\) that vanishes only at 0 and \(\Phi (t)/t\rightarrow 0\) as \(t\rightarrow 0\) and \(\Phi (t)/t\rightarrow \infty \) as \(t\rightarrow \infty \). By \(\Phi ^*\) we denote the Young function complementary to \(\Phi \) in the sense of Young.

We say that a Young function \(\Psi \) *increases more rapidly* than \(\Phi \) (in symbols, \(\Phi \prec \Psi \)) if for an arbitrary \(c>0\) there exists \(d>0\) such that \(c\Phi (t)\le \frac{1}{d}\Psi (dt)\) for all \(t\ge 0\).

If \(\Phi \prec \Psi \), then \(L^\Psi \subset L^\Phi \) and by \(i_\Psi :L^\Psi \rightarrow L^\Phi \) we denote the inclusion map.

### Proposition 2.3

Let \(T:L^\Phi \rightarrow X\) be a Bochner representable operator. Then for every Young function \(\Psi \) with \(\Phi \prec \Psi \) the operator \(T\circ i_\Psi :L^\Psi \rightarrow X\) is compact.

### Proof

Assume that \(\Psi \) is a Young function with \(\Phi \prec \Psi \). Then by [16, Theorem 5.3.3, p. 171] the closed unit ball \(B_\Psi (1)\) in \(L^\Psi \) is relatively \(\sigma (L^\Phi ,L^{\Phi ^*})\)-compact in \(L^\Phi \). Hence according to Theorem 2.1 \(T(B_\Psi (1))\) is relatively norm compact, and this means that \(T\circ i_\Psi :L^\Psi \rightarrow X\) is compact. \(\square \)

We say that a Young function \(\Phi \) *increases essentially more rapidly than another* \(\Psi \) (in symbols, \(\Psi \ll \Phi \)) if for an arbitrary \(c>0\), \(\Psi (ct)/\Phi (t)\rightarrow 0\) as \(t\rightarrow 0\) and \(t\rightarrow \infty \).

The following characterization of the mixed topology \(\gamma _\Phi (=\gamma [\mathcal{T}_\Phi ,\mathcal{T}_0])\) on \(L^\Phi \) will be useful (see [14, Theorem 2.1]).

### Theorem 2.4

Let \(\Phi \) be a Young function. Then the mixed topology \(\gamma _\Phi \) on \(L^\Phi \) is generated by the family of norms \(\{\Vert \cdot \Vert _\Psi \big |_{L^\Phi }:\Psi \ll \Phi \}\).

As a consequence of Theorems 2.2 and 2.4 we have:

### Corollary 2.5

Assume that a Young function \(\Phi ^*\) satisfies the \(\Delta _2\)-condition. Let \(T:L^\Phi \rightarrow X\) be a Bochner representable operator. Then there exists a Young function \(\Psi \) with \(\Psi \ll \Phi \) such that \(T(B_\Psi (1)\cap L^\Phi )\) is a relatively norm compact set in *X*.

### Remark

(i) The result of Corollary 2.5 was established in a different way in [14, Theorem 2.3].

(ii) For a bounded linear operator \(T:L^\Phi \rightarrow X\), following [6] one can define its functional norm |||*T*||| by \(|||T|||=\sup \Sigma \Vert \alpha _iT(\mathbb {1}_{A_i})\Vert _X,\) where the supremum is taken over all finite \(\Sigma \)-partition \((A_i)\) of \(\Omega \) and all \(\alpha _i\in \mathbb {R}\) such that \(\Vert \Sigma \alpha _i\mathbb {1}_{A_i}\Vert _\Phi \le 1\).

Uhl [18, Theorem 1] showed that if *X* either is reflexive or is a separable dual Banach space and \(\Phi \) obeys the \(\Delta _2\)-condition, then every bounded linear operator \(T:L^\Phi \rightarrow X\) with \(|||T|||<\infty \) is Bochner representable. If, in addition, \(\Phi ^*\) also obeys the \(\Delta _2\)-condition, then every bounded operator \(T:L^\Phi \rightarrow X\) with \(|||T|||<\infty \) is compact (see [18, Corollary 2]).

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