A remark on the approximation of \(p\)-compact operators by finite-rank operators
type: Item
instance of: scholarly article
title: A remark on the approximation of \(p\)-compact operators by finite-rank operators
zbMATH Open document ID: 1241.46013
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-01-04T00:00:00Z
review text: Let \(X\) and \(Y\) be Banach spaces, \(p \geq 1\), and \(q\) the conjugate index of \(p\), i.e., \(1/p + 1/q = 1\). A linear operator \(T:Y \to X\) is said to be \(p\)-compact if the image \(T(B_Y)\) of the closed unit ball \(B_Y\) is contained in \(\{\sum_n a_nx_n: (a_n) \in B_{\ell_q}\}\) (\((a_n) \in B_{c_0}\) if \(p = 1\)) for some \(X\)-valued absolutely \(p\)-summable sequence \((x_n) \in \ell_p (X)\). The \(\|\cdot\|_{\mathcal K_p}\)-norm of such an operator \(T\) is given by \(\|T\|_{\mathcal K_p} = \inf \|(x_n)\|_p\) where the infimum is taken over all \((x_n) \in \ell_p(X)\). Note that the space \(\mathcal K_p(Y,X)\) of all \(p\)-compact operators contains \(\mathcal F(Y,X)\), the space of finite-rank operators from \(Y\) to \(X\), and is moreover a Banach operator ideal. Recall that \(X\) has the approximation property (\(AP\)) if the identity operator can be approximated by finite-rank operators \(\mathcal F(X,X)\) uniformly on compact subsets of \(X\). The following two successively weaker companions of the \(AP\) have also been studied: (i) \(X\) as the \((p,p)\)-approximation property \(((p,p)\)-\(AP)\) if for every sequence \((x_n) \in \ell_p(X)\) and for every \(\varepsilon >0\) there exists \(S \in \mathcal F(X,X)\) such that \(\|(x_n - Sx_n)\|_p < \varepsilon\). (ii) \(X\) has the \(\mathcal K_p\)-approximation property \((\mathcal K_p\)-\(AP)\) if for every Banach space \(Y\) the space \(\mathcal F(Y,X)\) is \(\|\cdot\|_{\mathcal K_p}\)-dense in \(\mathcal K_p(Y,X)\). The author proves (Proposition \(3\)) that a Banach space has the \((p,p)\)-\(AP\) if and only if it has the \(AP\). In particular, by replacing the \((p,p)\)-\(AP\) with the \(AP\), this improves a result in [\textit{J. M. Delgado}, \textit{C. Piñeiro}, and \textit{E. Serrano}, ``Density of finite rank operators in the Banach space of \(p\)-compact operators'', J. Math. Anal. Appl. 370, No. 2, 498--505 (2010; Zbl 1205.46010)] saying that for a positive measure \(\mu\) closed subsp…
reviewed by: Trond Arnold Abrahamsen
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47L20
zbMATH DE Number: 5994355
zbMATH Keywords: approximation property defined by a Banach operator ideal
zbMATH Keywords: \(p\)-compact operators
zbMATH Keywords: \(L_p\)-spaces
zbMATH Keywords: Saphar's approximation property
author: Eve Oja
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.09.055
OpenAlex ID: W2018766486
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DOI: 10.1016/J.JMAA.2011.09.055
description: scientific article; zbMATH DE number 5994355