COMPLETELY CONTINUOUS SUBSPACES OF OPERATOR IDEALS

Abstract

Ülger, Saksman and Tylli have shown that if $X$ is a reflexive Banach space and $\mathcal{A}$ is a subalgebra of $K(X)$ such that $\mathcal{A}^*$ has the Schur property, then $\mathcal{A}$ is completely continuous. Here by introducing the concept of a strongly completely continuous subspace of an operator ideal, we improve their results. In particular, when $X$ is an $l_p$- direct sum and $Y$ is an $l_q$- direct sum of finite-dimensional Banach spaces with $1 \lt p \leq q \lt \infty$, we give a characterization of Schur property of the dual $\mathcal{M}^*$ of a closed subspace $\mathcal{M} \subseteq K(X,Y)$ in terms of strong complete continuity of $\mathcal{M}$.