Commutators on \((\sum\ell_q)_{\ell_1}\)
type: Item
instance of: scholarly article
title: Commutators on \((\sum\ell_q)_{\ell_1}\)
zbMATH Open document ID: 1343.47047
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: Let \(X, Y, Z\) be Banach spaces. A bounded linear operator \(T: X \to Y\) is said to be \(Z\)-strictly singular if there is no subspace \(Z_0\) of \(X\) which is isomorphic to \(Z\) and such that the restriction map \(T \mid_{Z_0}\) is an isomorphism. In the present paper, the author considers the case where \(Y=X= (\sum l_q)_{l_1}\) with \(1 \leq q < \infty\). Firstly, he shows that the unique maximal ideal in \(X\) is the set of all \(X\)-strictly singular operators. Secondly, he shows that \(T\) is a commutator if and only if for every nonzero \(\lambda \in \mathbb C\), \(T-\lambda I\) is not \(X\)-strictly singular. It should be pointed out that this paper is a continuation and extension of earlier work (see [\textit{D.-Y. Chen} et al., Stud. Math. 206, No. 2, 175--190 (2011; Zbl 1232.47031); corrigendum ibid. 223, No. 2, 187--191 (2014; Zbl 1302.47054)]).
reviewed by: Nadia Boudi
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46B45
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 6420185
zbMATH Keywords: commutators
zbMATH Keywords: maximal ideal
zbMATH Keywords: strictly singular operators
zbMATH Keywords: complemented subspaces
author: Bentuo Zheng
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.11.066
OpenAlex ID: W48250015
cites work: Q5660222
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cites work: Q4136031
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DOI: 10.1016/J.JMAA.2013.11.066
description: scientific article; zbMATH DE number 6420185