The \(p\)-approximation property in terms of density of finite rank operators
type: Item
instance of: scholarly article
title: The \(p\)-approximation property in terms of density of finite rank operators
zbMATH Open document ID: 1168.46008
author: J. Martínez
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-13T00:00:00Z
review text: The paper deals with the concept of \(p\)-approximation property, due to \textit{D.\,P.\thinspace Sinha} and \textit{A.\,K.\thinspace Karn} [Stud.\ Math.\ 150, No.\,1, 17--33 (2002; Zbl 1008.46008)]; this notion is a natural extension of the well-known approximation property in the context of Banach space theory. Let \(1\leq p\leq\infty\) and \(p^{\prime}\) be the conjugate of \(p\). The \(p\)-convex hull of a sequence \((x_{n})\in l_{p}(X)\) is the set \[ p-\text{conv}\{x_{n}\}=\left\{{\displaystyle\sum\limits_{n=1}^{\infty}}a_{n}x_{n};{\displaystyle\sum\limits_{n=1}^{\infty}}\left| a_{n}\right| ^{p^{\prime}}\leq1\right\}. \] If \(p=1\), the second sum is naturally replaced by \(\sup\left| a_{n}\right|\). A set \(K\subset X\) is relatively \(p\)-compact if there exists \((x_{n})\in l_{p}(X)\) (\((x_{n})\in c_{0}(X)\) if \(p=\infty\)) such that \(K\subset p-\text{conv}\{x_{n}\}\). A Banach space \(X\) has the \(p\)-approximation property (\(p\)-\(AP\)) if the identity \(I_{X}\) can be approximated by finite rank operators uniformly on the relatively \(p\)-compact subsets of \(X\). The ideals of bounded, nuclear, compact, \(p\)-summing and finite rank operators are denoted by \(\mathcal{L},\mathcal{N},\mathcal{K},\mathcal{P}_{p}\) and \(\mathcal{F}\), respectively. If \(\mathcal{A}\) is an operator ideal, \(\mathcal{A}^{d}\) denotes its dual operator ideal. An operator \(T\in\mathcal{L}(X,Y)\) is\ \(p\)-compact if \(T(B_{X})\) is relatively \(p\)-compact in \(Y\); the set of all \(p\)-compact operators from \(X\) to \(Y\) is denoted by \(\mathcal{K}_{p}(X,Y).\) The main result of the paper characterizes the \(p\)-\(AP\) in terms of density of finite rank operators in the spaces of \(p\)-compact operators. More precisely, if \(X\) is a Banach space and \(1\leq p\leq\infty\), the authors show that the following statements are equivalent: (a) \(X\) has the \(p\)-\(AP\). (b) \(\mathcal{F}(Y,X)\) is \(\left\|\cdot\right\| \)-dense in \(\mathcal{K}_{p}(Y,X)\) for every Ban…
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 5553661
zbMATH Keywords: \(p\)-approximation property
zbMATH Keywords: \(p\)-compact operator
zbMATH Keywords: \(p\)-summing operator
zbMATH Keywords: quasi-\(p\)-nuclear operator
zbMATH Keywords: \(p\)-nuclear operator
zbMATH Keywords: dual operator ideal
reviewed by: Daniel Pellegrino
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.12.047
OpenAlex ID: W2128423703
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DOI: 10.1016/J.JMAA.2008.12.047
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description: scientific article; zbMATH DE number 5553661