The dual space of $(L(X,Y), \tau_p)$ and the p-approximation property

Abstract

We establish a representation of the dual space of L(X, Y), the space of bounded linear operators from a Banach space X into a Banach space Y. endowed with the topology tau(p) of uniform convergence on p-compact subsets of X We apply this representation and solve the duality problem for the p-approximation property (p-AP), that is, if the dual space X* has the p-AP, then so does X However, the converse does not hold in general. We show that given 2 < p < infinity, there exists a subspace of I-q which fails to have the p-AP, when q >2p/(p - 2). This subspace is the Davie space in I-q (Davie (1973) [5]) which does not have the approximation property. It follows that for every 2 < p < infinity there exists a Banach space Y-p such that it has the p-AP, but its dual space Y-p* fails to have the p-AP We study the relation of the p-AP with the denseness of finite rank operators in the topology tau(p) Finally we introduce the p-compact approximation property (p-CAP) and show for every 2 < p < infinity that the Davie space in c(0) fails to have the p-CAP, and also that a variant of the Willis space (Willis (1992) [17]) has the p-CAP, but it fails to have the p-AP. (C) 2010 Elsevier Inc All rights reserved.