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.

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, that is, 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:

$$\begin{aligned} E':=\bigg \{v\in L^0:\int _\Omega |u(\omega )v(\omega )|\,d\mu <\infty \ \text{ for } \text{ all } u\in E\bigg \}. \end{aligned}$$

The associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is defined for \(v\in E'\) by

$$\begin{aligned} \Vert v\Vert _{E'}:=\sup \left\{ \Big |\int _\Omega u(\omega )v(\omega )\,d\mu \Big |:u\in E, \Vert u\Vert _E\le 1\right\} . \end{aligned}$$

Then \(\mathrm{supp}\,E'=\Omega \) (see [9, Theorem 6.1.5]). The \(\sigma \)-order continuous dual \(E^\sim _c\) of 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

$$\begin{aligned} F_v(u)=\int _\Omega u(\omega )v(\omega )\,d\mu \ \text{ for } u\in E \end{aligned}$$

(see [9, Theorem 6.1.1]). The Mackey topology \(\tau (E,E')\) is a locally convex-solid Hausdorff topology with the \(\sigma \)-Lebesgue property (see [1, Exercise 18, p. 178]). Note that \(\tau (E,E')=\mathcal{T}_E\) if and only if \(\Vert \cdot \Vert _E\) is \(\sigma \)-order continuous.

In addition to these facts, the following assumptions will be in force without futher mention throughout this paper:

  1. (A)

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

  2. (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]).

Note that from (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,

$$\begin{aligned} (E')_a:=\left\{ v\in E':\Vert v_n\Vert _{E'}\rightarrow 0 \ \text{ if } \ |v(\omega )|\ge v_n(\omega ) \downarrow 0 \ \mu -a.e.\right\} . \end{aligned}$$

Then \(\Vert \cdot \Vert _{E'}\) is order continuous if and only if \((E')_a=E'\).

Let \(L^0(X)\) stand for the linear space of \(\mu \)-equivalence classes of all strongly \(\Sigma \)-measurable functions \(g:\Omega \rightarrow X\). Let

$$\begin{aligned} E'(X)=\left\{ g\in L^0(X):\Vert g(\cdot )\Vert _X\in E'\right\} . \end{aligned}$$

Definition 1.1

A bounded linear operator \(T:E\rightarrow X\) is said to be Bochner representable, if there exists \(g\in E'(X)\) such that

$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

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

Assume that \(T:E\rightarrow X\) is a Bochner representable operator. Then the following statements hold:

  1. (i)

    T is \((\tau (E,E'),\Vert \cdot \Vert _X)\)-continuous.

  2. (ii)

    T maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X.

Proof

There exists \(g\in E'(X)\) such that

$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

(i) Assume that \(x^*\in X^*\). Then for \(u\in E\),

$$\begin{aligned} \displaystyle |x^*(T(u))|= & {} \bigg |\int _\Omega u(\omega )x^*(g(\omega ))\,d\mu \bigg |\\\le & {} \int _\Omega |u(\omega )|\cdot \Vert x^*\Vert _{X^*} \Vert g(\omega )\Vert _X\, d\mu = \Vert x^*\Vert _{X^*} F_{\Vert g(\cdot )\Vert _X}(|u|). \end{aligned}$$

It follows that \(x^*\circ T\in E^\sim _c\), so 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]).

(ii) Assume that 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

$$\begin{aligned}&\sup _{u\in H}\int _{\Omega \smallsetminus \Omega _0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}&\sup _{u\in H}\int _A |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \ \ \text{ whenever } \ \ A\in \Sigma \ \text{ and } \ \mu (A)\le \delta . \end{aligned}$$
(2.2)

Choose a sequence \((s_n)\) of 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

$$\begin{aligned} T_n(u):=\int _{A_0} u(\omega )s_n(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

Note that for every \(n\in \mathbb {N}\), \(T_n\) is a bounded operator of finite dimensional range, so that \(T_n\) is a compact operator. Define an operator \(T_{A_0}:E\rightarrow X\) by

$$\begin{aligned} T_{A_0}(u):=\int _{A_0} u(\omega )g(\omega )\,d\mu \ \ for \ \ u\in E. \end{aligned}$$

Then for \(u\in B_E(1)\), we have

$$\begin{aligned} \displaystyle \bigg \Vert \int _{A_0} u(\omega )(g(\omega )-s_n(\omega ))\,d\mu \bigg \Vert _X\le & {} \int _{A_0}|u(\omega )|\cdot \Vert g(\omega )-s_n(\omega )\Vert _X\,d\mu \\\le & {} \, \bigg (\int _\Omega |u(\omega )|\mathbb {1}_{A_0}(\omega )\,d\mu \bigg ) \sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X\\\le & {} \;\big \Vert \mathbb {1}_{A_0}\big \Vert _{E'}\sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X. \end{aligned}$$

Hence \(\Vert T_{A_0}-T_n\Vert \le \Vert \mathbb {1}_{A_0}\Vert _{E'}\sup _{\omega \in A_0}\Vert g(\omega )-s_n(\omega )\Vert _X\), and hence \(\Vert T_{A_0}-T_n\Vert \rightarrow 0\). It follows that \(T_{A_0}\) is a compact operator. Since \(\sup _{u\in H}\Vert u\Vert _E<\infty \), we have that \(K_\varepsilon :=\{T(\mathbb {1}_{A_0} u):u\in H\}\) is a relatively norm compact set in X.

Moreover, since \(\mu (\Omega _0\smallsetminus A_0)\le \delta \), for every \(u\in H\), by (2.2) we get

$$\begin{aligned} \big \Vert T\big (\mathbb {1}_{\Omega _0\smallsetminus A_0}u\big )\big \Vert _X\le \int _{\Omega \smallsetminus A_0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2} \end{aligned}$$

i.e., \(\{T(\mathbb {1}_{\Omega _0\smallsetminus A_0}u):u\in H\}\subset \frac{\varepsilon }{2} B_X\). For every \(u\in H\), by (2.1) we have

$$\begin{aligned} \big \Vert T\big (\mathbb {1}_{\Omega \smallsetminus \Omega _0}u\big )\big \Vert _X\le \int _{\Omega \smallsetminus \Omega _0} |u(\omega )|\cdot \Vert g(\omega )\Vert _X \,d\mu \le \frac{\varepsilon }{2}, \end{aligned}$$

i.e., \(\{T(\mathbb {1}_{\Omega \smallsetminus \Omega _0}u):u\in H\}\subset \frac{\varepsilon }{2} B_X\). Hence for every \(u\in H\), we have

$$\begin{aligned} T(u)=T(\mathbb {1}_{\Omega \smallsetminus \Omega _0}u)+T(\mathbb {1}_{\Omega _0\smallsetminus A_0} u)+ T(\mathbb {1}_{A_0} u)\in \varepsilon B_X+K_\varepsilon . \end{aligned}$$

In view of the Grothendieck’s compactness criterion (see [4, Exercise 4(iii)]) T(H) is a relatively norm compact set in X, as desired. \(\square \)

By \(\gamma [\mathcal{T}_E,\mathcal{T}_0]\) (in brief, \(\gamma _E\)) we denote the 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:

$$\begin{aligned} (E,\gamma _E)^*=\{F_v:v\in (E')_a\}. \end{aligned}$$
(2.3)

Theorem 2.2

Assume that \((E,\Vert \cdot \Vert _E)\) is a Banach function space with the order continuous accociated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\). Let \(T:E\rightarrow X\) be a Bochner representable operator. Then the following statements hold:

  1. (i)

    T is \((\gamma _E,\Vert \cdot \Vert _X)\)-continuous and norm-compact operator.

  2. (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

(i) There exists \(g\in (E')_a(X)\) such that

$$\begin{aligned} T(u)=\int _\Omega u(\omega )g(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

For \(u\in E\), we have

$$\begin{aligned} \Vert T(u)\Vert _X\le \int _\Omega |u(\omega )|\;\Vert g(\omega )\Vert _X \,d\mu =F_{\Vert g(\cdot )\Vert _X}(|u|), \end{aligned}$$

where \(\Vert g(\cdot )\Vert _X\in (E')_a\). Using (2.3) we obtain that 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 \(\Vert \;\Vert s_n(\cdot )-g(\cdot )\Vert _X \;\Vert _{E'}\le \Vert v_n\Vert _{E'}\rightarrow 0\).

For each \(n\in \mathbb {N}\), let \(T_n:E\rightarrow X\) be a linear operator defined by

$$\begin{aligned} T_n(u):=\int _\Omega u(\omega )s_n(\omega )\,d\mu \ \ \text{ for } \ \ u\in E. \end{aligned}$$

Note that the range of each \(T_n\) is contained in the span of the finite set of values of \(s_n\). Therefore \(T_n\) is compact for each \(n\in \mathbb {N}\), and for each \(u\in E\),

$$\begin{aligned} \displaystyle \Vert (T_n-T)(u)\Vert _X= & {} \,\bigg \Vert \int _{\Omega }u(\omega )(s_n(\omega )-g(\omega ))\,d\mu \bigg \Vert _X\\\le & {} \int _\Omega |u(\omega )|\cdot \Vert s_n(\omega )-g(\omega )\Vert _X d\,\mu \\\le & {} \,\Vert u\Vert _E \cdot \Big \Vert \,\Vert s_n(\cdot )-g(\cdot )\Vert _X\,\Big \Vert _{E'}. \end{aligned}$$

It follows that \(\Vert T-T_n\Vert \rightarrow 0\), so 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.

The Orlicz space

$$\begin{aligned} L^\Phi :=\bigg \{u\in L^0:\int _\Omega \Phi (\lambda |u(\omega )|)\,d\mu <\infty \ \ \text{ for } \text{ some } \ \ \lambda >0\bigg \}, \end{aligned}$$

equipped with the topology \(\mathcal{T}_\Phi \) of the norm:

$$\begin{aligned} \Vert u\Vert _\Phi :=\inf \bigg \{\lambda >0:\int _\Omega \Phi (|u(\omega )|/\lambda )\,d\mu \le 1\bigg \} \end{aligned}$$

is a perfect Banach function space (see [10, 16]). Then \((L^\Phi )'=L^{\Phi ^*}\) and \((L^{\Phi ^*})_a=E^{\Phi ^*}=\{v\in L^{\Phi ^*}: \int _\Omega \Phi (\lambda |v(\omega )|)d\mu <\infty \) for all \(\lambda >0\}\). In particular, \(E^{\Phi ^*}=L^{\Phi ^*}\) if \(\Phi ^*\) satisfies the \(\Delta _2\)-condition, i.e., \(\Phi ^*(2t)\le d\Phi ^*(t)\) for some \(d>1\) and all \(t\ge 0\).

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]).