Operators on the space of bounded strongly measurable functions
type: Item
instance of: scholarly article
title: Operators on the space of bounded strongly measurable functions
zbMATH Open document ID: 1248.47035
author: Marian Nowak
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-11T00:00:00Z
review text: Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(\Omega\), \(X,Y\) be Banach spaces, and \(\mathcal L^\infty(\Sigma,X)\) be the space of all strongly measurable bounded functions \(f: \Omega \to X\) equipped with the supremum norm. A bounded linear operator \(T: \mathcal L^\infty(\Sigma,X) \to Y\) is called \(\sigma\)-smooth if it maps uniformly bounded sequences pointwise convergent to zero to norm convergent ones. The paper consists of four sections. The first one contains introduction and terminology. In the second one, a characterization of \(\sigma\)-smooth operators \(T: \mathcal L^\infty(\Sigma,X) \to Y\) in terms of their representing measure is given and the following Banach-Steinhaus type theorem for \(\sigma\)-smooth operators is proved: if a sequence of \(\sigma\)-smooth operators \(T_n: \mathcal L^\infty(\Sigma,X) \to Y\) converges pointwise on \(L^\infty(\Sigma,X)\), then this sequence is uniformly \(\sigma\)-smooth, and the limiting operator is also \(\sigma\)-smooth. The third section deals with the space of all \(\sigma\)-smooth functionals on \(\mathcal L^\infty(\Sigma,X)\), and in the fourth section two sufficient conditions for an operator \(T: \mathcal L^\infty(\Sigma,X) \to Y\) to be \(\sigma\)-smooth are established.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 6005697
zbMATH Keywords: strongly measurable functions
zbMATH Keywords: operator-valued measures
zbMATH Keywords: \(\sigma \)-smooth operators
zbMATH Keywords: weakly compact operators
zbMATH Keywords: weak\(^{*}\) sequential compactness
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.10.033
OpenAlex ID: W2037915076
cites work: Weak precompactness, strong boundedness, and weak complete continuity
cites work: Q4267848
cites work: Invariant Subspaces for Positive Operators Acting on a Banach Space with Basis
cites work: Q4438875
cites work: Q5584192
cites work: Q4402591
cites work: Weak compactness in the space of vector measures
cites work: Q3329946
cites work: Q4148368
cites work: Q4519895
cites work: On integration in Banach spaces, I
cites work: Q3254057
cites work: Integral representation of linear operators on Orlicz-Bochner spaces
cites work: On Banach spaces whose dual balls are not weak\(^*\) sequentially compact
cites work: Q3943431
cites work: Q5590337
cites work: Q5653224
cites work: Q4452867
cites work: Q5801558
cites work: Q5528917
cites work: A Grothendieck-type theorem for the space of totally measurable functions
cites work: Conditional weak compactness and weak sequential completeness in vector-valued function spaces
cites work: A generalized Orlicz-Pettis theorem and applications
cites work: Q3946374
cites work: A variant of Grothendieck's theorem on weak* convergent sequences
cites work: On weak compactness in spaces of measures
DOI: 10.1016/J.JMAA.2011.10.033
description: scientific article; zbMATH DE number 6005697