Projections in the spaces of bounded linear operations

Tsang Hai Kuo

Vol. 52 (1974), No. 2, 475–480

DOI: 10.2140/pjm.1974.52.475

Abstract

For Banach spaces X,Z, let B(X,Z) denote the space of bounded linear operators from X into Z and K(X,Z) (resp. W(X,Z)) the subspace of compact (resp. weakly compact) operators. It is shown that (a) if X contains an isomorph of c₀, then K(X,l^∞) is not complemented in B(X,l^∞), (b) if S is a compact Hausdorff space which is not scattered, then K(C(S),Z) is not complemented in W(C(S),Z) for Z = c₀ or l^∞. In particular, K(l^∞,c₀) is not complemented in B(l^∞,c₀), which gives a negative answer to a question proposed by Arterburn and Whitley.

Mathematical Subject Classification

Primary: 46B05

Secondary: 47D15

Milestones

Received: 15 August 1973

Revised: 21 January 1974

Published: 1 June 1974