Compact-friendly multiplication operators on Banach function spaces
type: Item
instance of: scholarly article
title: Compact-friendly multiplication operators on Banach function spaces
zbMATH Open document ID: 1034.47011
author: Gleb Sirotkin
published in: Journal of Functional Analysis
publication date: 2002-11-28T00:00:00Z
review text: It was proved by \textit{Y. A. Abramovich} et al. [Indag. Math., New Ser. 10, 161--171 (1999; Zbl 1043.47504)], that a multiplication operator on each \(L_p(\mu)\)-space or \(C(K)\) space is compact-friendly if and only if the multiplier is constant on a set of positive measure. The proof depended heavily on the metric properties of the \(L_p\) and \(C(K)\) spaces. Answering a question posed in that article, the author shows in the present paper that the metric properties of the spaces involved are not essential and that the result remains true for each Banach function space. The main result of the paper under review is as follows: Let \((\Omega,\Sigma,\mu)\) be an arbitrary measure space without infinite atoms. Consider a Banach function space \(X\) over \(\Omega\). Then for any \(0\leq h\in L_\infty(\Omega)\) the multiplication operator \(M_h\in L(X)\) is compact-friendly if and only if \(h\) is constant on a set of positive measure.
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1836666
zbMATH Keywords: flat
zbMATH Keywords: positive operator
zbMATH Keywords: measure space
zbMATH Keywords: Banach function space
zbMATH Keywords: multiplication operator
zbMATH Keywords: compact-friendly
reviewed by: Bahmann Yousefi
MaRDI profile type: Publication
full work available at URL: jfan.2001.3931
OpenAlex ID: W2080481109
cites work: Invariant subspace theorems for positive operators
cites work: Multiplication and compact-friendly operators
cites work: Q4243021
cites work: A characterization of compact-friendly multiplication operators.
cites work: Q3683058
cites work: Q4136031
DOI: 10.1006/JFAN.2001.3931
description: scientific article; zbMATH DE number 1836666