Matrix Maps and the Isomorphic Structure of BK Spaces

Abstract

This note gives a characterization of BK spaces that contain isomorphic copies of ${c_0}$ in terms of matrix maps and a sufficient condition for a matrix map from ${l_\infty }$ into a BK space to be a compact operator. The primary tool used in this note is the Bessaga-Pelczynski characterization of Banach spaces which contain isomorphic copies of ${c_0}$. It is shown that weakly compact matrix maps on ${l_\infty }$ are compact and that, if $E$ is a BK space such that there is a matrix $A$ such that ${c_0} \subseteq {E_A}$ and ${E_A}$ is not strongly conull, then $E$ must contain an isomorphic copy of ${c_0}$.