Operators preserving disjointness on rearrangement invariant spaces
type: Item
instance of: scholarly article
title: Operators preserving disjointness on rearrangement invariant spaces
zbMATH Open document ID: 0689.47012
author: Y. A. Abramovich
published in: Pacific Journal of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: Let X and Y be two rearrangement invariant spaces on a measure space (\(\Omega\),\(\Sigma\),\(\mu)\) with a finite, nonatomic measure \(\mu\). We show that if there exists a non-zero order continuous disjointness preserving operator T:X\(\to Y\), then \(X\subseteq Y\). [Let us remark incedentally that the proof depends on the multiplicative representation of disjointness preserving operators obtained by the author in Indag. Math. 45, 265-279 (1983; Zbl 0527.47025). This result has many corollaries of which we mention two: 1) If \(T:L_ p(\Omega,\Sigma,\mu)\to L_ q(\Omega,\Sigma,\mu)\) \((0<p<q\leq \infty)\) preserves disjointness, then \(T\equiv 0\). 2) If X and Y are rearrangement invariant Banach function spaces with strictly monotone norms and T is a positive isometry from X onto Y, then \(X=Y\).
reviewed by: Y. A. Abramovich
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4129476
zbMATH Keywords: rearrangement invariant spaces
zbMATH Keywords: nonatomic measure
zbMATH Keywords: order continuous disjointness preserving operator
MaRDI profile type: Publication
full work available at URL: pjm.1991.148.201
OpenAlex ID: W2000361076
DOI: 10.2140/PJM.1991.148.201
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description: scientific article; zbMATH DE number 4129476