On positive strictly singular operators and domination
type: Item
instance of: scholarly article
title: On positive strictly singular operators and domination
zbMATH Open document ID: 1063.47028
DOI: 10.1023/A:1025836620741
author: Julio Flores
author: Francisco L. Hernández
published in: Positivity
publication date: 2003-11-10T00:00:00Z
review text: Let \(X, Y\) be Banach spaces. An operator \(T: X \rightarrow Y\) is called strictly singular if and only if every infinite-dimensional subspace \(M\) of \(X\) contains an infinite-dimensional subspace \(N\) such that the restriction of \(T\) to \(N\) is compact. The authors study the problem of domination by strictly singular operators. In some very special cases, the problem has an affirmative answer. They provide examples showing that, in general, the problem of domination for strictly singular operators has a negative answer. The main contribution is the following result. Theorem: Let \(E\) and \(F\) be order continuous Banach lattices such that \(E\) has the subsequence splitting property and the dual norm on \(E'\) is order continuous. If \(T\) is a positive strictly singular operator from \(E\) to \(F\), then every operator in \([0,T]\) is strictly singular. The authors provide an example to show that the order continuity of \(F\) is indispensible.
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 2002005
zbMATH Keywords: strictly singular operator
zbMATH Keywords: domination
zbMATH Keywords: order continuous Banach lattices
reviewed by: Şafak Alpay
MaRDI profile type: Publication
full work available at URL: a:1025836620741
OpenAlex ID: W205621352
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description: scientific article; zbMATH DE number 2002005