On the best compact approximation problem for operators between \(L_ p\)- spaces
type: Item
instance of: scholarly article
title: On the best compact approximation problem for operators between \(L_ p\)- spaces
zbMATH Open document ID: 0642.41016
DOI: 10.1016/0021-9045(87)90040-2
author: H. Bang
author: E. Odell
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: A pair of Banach spaces (X,Y) is said to have the best compact approximation property (b.c.a.p.) if, for every bounded linear operator \(T: X\to Y\) there exists a compact operator \(K: X\to Y\) satisfying \(\| T-K\| \leq \| T-K'\|\) for all compact K'. We say that X has the b.c.a.p. if the pair (X,X) has this property. It is known that \(\ell_ p\) has the b.c.a.p. for \(1\leq p<\infty\) while \(\ell_{\infty}\), \(L_ 1\), \(L_{\infty}\) and C[0,1] fail to have it. \textit{V. Benyamini} and \textit{P. K. Lin} [Israel J. Math. 51, 298-304 (1985; Zbl 0596.47020)] showed that for \(1<p<\infty\), \(p\neq 2\), Lp fails the best compact approximation property. In this paper this result is extended by showing that \((\ell_ p,L_ p)\) fails the b.c.a.p. for \(1<p<2\). A sufficient condition is also given for an operator \(T: L_ p\to L_ p\) \((2<p<\infty)\) to have a best compact approximant (b.c.a.). Main results: 1. Let \(1<p<2\). Then \((\ell_ p,L_ p)\) fails the b.c.a.p. 2. Let \(2<p<p\) and let \(T: L_ p\to L_ p\) be a bounded linear operator satisfying: If \((x_ n)\) is a uniformly bounded weakly null sequence in \(L_ p\), then \(\| Tx_ n\| \to 0\) then T has a best compact approximant.
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4046324
zbMATH Keywords: best compact approximation property
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90040-2
OpenAlex ID: W1966954662
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description: scientific article; zbMATH DE number 4046324