English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Given a subspace X of lp, 1 < p < oo, the compact operators on X are an M-ideal in the bounded linear operators on X if and only if X has the compact approximation property. 0. Introduction. Recently Harmand and Lima [7] proved that if A" is a Banach space for which K(X), the space of compact operators on X, is an M-ideal in L(X), the space of bounded linear operators on X, then there is a net {7a} in K(X) so that: (i) Ta -» 7 strongly, '(ii)||7a||^lforalla, (iii) T* -* I strongly, (iv) lim.H/r„|| = l. The main result of this paper is a strong converse to the Harmand-Lima theorem for subspaces of lp, 1 < p < oo. In Theorem 6 we show that if X is a subspace of (LXn)p (dim A'„<oo;l</'<oo) which has the compact approximation property, then K(X) is an M-ideal in L( X). Part of the proof consists in showing that such an X satisfies conditions (i)-(iv) in the Harmand-Lima theorem. This result (which is simple given the state-of-the-art in Banach space theory) is proved for general reflexive spaces in §2. §3 is devoted to proving the converse of the Harmand-Lima theorem for subspaces of (LXn)p. Here we use blocking methods which have been previously used in the study of isomorphic, rather than isometric, properties of lp and a few other spaces. 1. Notation and preliminaries. If X and Y are Banach spaces, L(X, Y) (resp. K(X, Y)) will denote the space of all bounded linear operators (resp. compact linear operators) from X to Y. If X = Y then we simply write L(X) (resp. K(X)). Ball(X) will denote the closed unit ball of X. Received by the editors April 2, 1984. 1980 Mathematics Subject Classification. Primary 46A32, 47B05; Secondary 47B05, 41A50.

A characterization of subspaces $X$ of $l\sb p$ for which $K(X)$ is an $M$-ideal in $L(X)$