Compactness of Bochner representable operators on Orlicz spaces
type: Item
instance of: scholarly article
title: Compactness of Bochner representable operators on Orlicz spaces
zbMATH Open document ID: 1172.47024
author: Marian Nowak
published in: Positivity
publication date: 2009-03-27T00:00:00Z
review text: Let \(L^\Phi\) be an Orlicz space over the \(\sigma\)-finite measure space \((\Omega,\Sigma,\mu)\) and \(X\) a Banach space. An operator \(T\) from \(L^\Phi\) to \(X\) is said to be Bochner representable if there is a function \(g\) belonging to the Orlicz--Bochner space \(L^{\Phi^*}(X)\) (where \(\Phi^*\) is the complementary function of \(\Phi\)) such that \(T(u)=\int_\Omega u(\omega) g(\omega) \, d \mu\), \(u \in L^\Phi\). In this paper, some results on compactness for this kind of operators are given. Starting from the original results of \textit{J.\,J.\thinspace Uhl jun.}\ [Stud.\ Math.\ 40, 17--22 (1971; Zbl 0228.47020)] for continuous operators with respect to the norm topologies, the results of the present paper extend them to the following version. Consider the mixed topology \(\gamma_{L^\Phi}\) on \(L^\Phi\) (the finest locally convex topology on \(L^\Phi\) which agrees with the topology of convergence in measure on \(\| \cdot\|_\Phi\)-bounded subsets). It is proved that every Bochner representable operator \(T:L^\Phi \to X\) is \((\gamma_{L^\Phi},\| \cdot \|_X)\)-compact (Theorem 2.3). As an application, an improvement of a particular case of the results of J.\,J.\thinspace Uhl quoted above is obtained; namely, every Bochner representable operator \(T:L^\infty \to X\) is \((\tau(L^\infty,L^1),\|\cdot\|)\)-compact (Theorem 3.3).
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5534422
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: mixed topologies
zbMATH Keywords: Mackey topologies
zbMATH Keywords: compact operators
zbMATH Keywords: Bochner representable operators
MaRDI profile type: Publication
full work available at URL: s11117-008-2179-x
OpenAlex ID: W2058235200
cites work: Q3254057
cites work: Orlicz spaces and modular spaces
cites work: Q3312762
cites work: Q3746322
cites work: Q4722591
cites work: Q4722592
cites work: Q3781572
cites work: Q5449085
cites work: Absolutely continuous and modularly continuous operators defined on spaces of measurable functions
cites work: On a class of operators on Orlicz spaces
cites work: Linear spaces with mixed topology
DOI: 10.1007/S11117-008-2179-X
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description: scientific article; zbMATH DE number 5534422