Pełczyński's property $V$ for spaces of compact operators
type: Item
instance of: scholarly article
title: Pełczyński's property $V$ for spaces of compact operators
zbMATH Open document ID: 1477.46018
author: Wuyi He
author: Lixin Cheng
published in: Positivity
publication date: 2021-08-16T00:00:00Z
review text: A bounded linear operator \(T : X \to Y\) between two Banach spaces is said to be unconditionally converging if for every unconditional Cauchy series \(\sum_n x_n\) in \(X\), the series \(\sum_n T x_n\) is unconditionally convergent. A Banach space \(X\) has Pełczyński's property (\(V\)) if for all Banach spaces \(Y\), every unconditionally converging operator \(T : X \to Y\) is weakly compact. In this paper the authors prove that if \(X\) is a reflexive Banach space with an unconditional basis, then the Banach space of compact operators on \(X\), \(K(X)\), has Pełczyński's property (\(V\)). In particular, \(K(L_p[0,1])\), for \(1 < p < \infty\), has Pełczyński's property (V). The proof uses a sequential representation of the space of compact operators on \(X\).
reviewed by: Vegard Lima
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47A80
zbMATH DE Number: 7382090
zbMATH Keywords: Pełczyński's property \((V)\)
zbMATH Keywords: compact operator
zbMATH Keywords: tensor product
zbMATH Keywords: reflexivity
zbMATH Keywords: Banach spaces
zbMATH Keywords: unconditional basis
MaRDI profile type: Publication
full work available at URL: s11117-020-00805-2
OpenAlex ID: W3125226996
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DOI: 10.1007/S11117-020-00805-2
description: scientific article; zbMATH DE number 7382090