New criteria of some bounded approximation properties
type: Item
instance of: scholarly article
title: New criteria of some bounded approximation properties
zbMATH Open document ID: 1247.46017
author: Ju Myung Kim
published in: Taiwanese Journal of Mathematics
publication date: 2011-12-19T00:00:00Z
review text: For Banach spaces \(X\) and \(Y\), let the Banach space of all bounded linear operators from \(X\) to \(Y\) be denoted by \({\mathcal L}(X,Y)\). Let \({\mathcal F}(X,Y)\) and \({\mathcal K}(X,Y)\) denote its subspaces of finite-rank and compact operators. A Banach space \(X\) is said to have the approximation property (AP) if the identity map \(I_X\) on \(X\) belongs to the closure of \({\mathcal F}(X,X)\) in the topology \(\tau\) of uniform convergence on compact subsets of \(X\). If \(I_X\) belongs to the \(\tau\)-closure of the subset of \({\mathcal F}(X,X)\) of the operators of norm \( \leq \lambda\), then \(X\) is said to have the \(\lambda\)-BAP. If \({\mathcal F}(X,X)\) is replaced by \({\mathcal K}(X,X)\), then the corresponding approximation properties are called the compact AP and the the compact \(\lambda\)-BAP, respectively. These are well-studied classical approximation properties. Now, if \({\mathcal F}(X,X)\) is replaced by an arbitrary linear subspace \({\mathcal A}(X,X)\) of \({\mathcal L}(X,X)\), then one obtains general versions of approximation properties, which have been studied since the 1980s by O. I. Reinov, N. Grønbæk, G. A. Willis, H.-O. Tylli, and others, including the reviewer. The author rediscovers these approximation properties (unfortunately, this has gone unnoticed by the referee) and names them the \({\mathcal A}\)-AP and the \(\lambda\)-\({\mathcal A}\)-BAP, respectively. From the main results under review, only Theorems 1.1 and 1.2 are essentially new. They state that \(X\) has the \(\lambda\)-\({\mathcal A}\)-BAP (respectively, the \({\mathcal A}\)-AP) if and only if, for every (separable reflexive) Banach space \(Y\), every \(T \in {\mathcal K}(X,Y)\) belongs to the \(\tau\)-closure of \(\{TS : S \in \mathcal A(X,X), \;\|S\| \leq \lambda\}\) (respectively, of \(\{TS : S \in \mathcal A(X,X)\}\)). The other results are known. For instance, Theorem 1.7 of the paper is contained in Corollary 1.4 of [\textit{O. I. Rejnov}, Math. Note…
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 5989926
zbMATH Keywords: approximation property
zbMATH Keywords: bounded approximation property
reviewed by: Eve Oja
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500406286
OpenAlex ID: W1546486938
DOI: 10.11650/TWJM/1500406286
description: scientific article; zbMATH DE number 5989926