Projections in the space of bounded linear operators

Abstract

Thorp has shown that for X and Y certain Banach spaces of sequences there is no continuous linear projection of the bounded linear operators from X to Y onto the compact linear operators from X to Y. In this paper, this result, as well as related results for the weakly compact linear operators, is demonstrated for cases including (a)X an infinite dimensional abstract L-space and Y an infinite dimensional space whose conjugate contains a countable total set and (b)X a separable >-space and Y = C(S) with S either a metric space containing an infinite number of points or S a compact space which contains a one-to-one convergent sequence.