Isometric properties of the subspace of compact operators in a space of continuous linear operators
type: Item
instance of: scholarly article
title: Isometric properties of the subspace of compact operators in a space of continuous linear operators
zbMATH Open document ID: 0694.47028
DOI: 10.1007/BF01158236
published in: Mathematical Notes
publication date: 1989-01-01T00:00:00Z
review text: The interesting problem of abstract functional analysis is the problem of Banach spaces E and F for which the space of all compact operators K(E,F) is a nontrivial M-ideal in the space L(E,F) of all bounded (continuous) linear operators. In the paper of Cho and Johnson it has been proved that \(K(E)=K(E,E)\) is an M-ideal in \(L(E)=L(E,E)\) if E is a closed subspace of \(\ell_ p\), \(1<p<\infty\), which satisfies a compact (metrical) approximation property. Some unsolved questions in the mentioned paper of Cho and Johnson are solved in the present paper. A new theorem is also proved in which the condition for a decomposition of the space L(E,F) into a direct sum of the annulator of K(E,F) and a subspace, which is isometrically isomorphic with K(E,F) is given. The consequence of this result is the effective sufficient condition for K(E,F) being the M-ideal in L(E,F). As a corollary the solution of Cho and Johnson's problems is obtained.
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 4136822
zbMATH Keywords: continuous linear operators
zbMATH Keywords: space of all compact operators
zbMATH Keywords: nontrivial M-ideal
zbMATH Keywords: approximation property
author: Eve Oja
MaRDI profile type: Publication
cites work: Q3695801
cites work: A Characterization of Subspaces X of l p for which K(X) is an M- Ideal in L(X)
cites work: Banach Spaces Which are M-Ideals in their Biduals
cites work: Q3915467
cites work: Q3684673
cites work: Q4136031
cites work: Remarks on Banach spaces of compact operators
full work available at URL: https://doi.org/10.1007/bf01158236
OpenAlex ID: W2164922546
description: scientific article; zbMATH DE number 4136822