On the Limited p-Schur Property of Some Operator Spaces

Abstract

We introduce and study the notion of limited $p$-Schur property ($1\leq p\leq\infty$) of Banach spaces. Also, we establish some necessary and sufficient conditions under which some operator spaces have the limited $p$-Schur property. In particular, we prove that if $X$ and $Y$ are two Banach spaces such that $X$ contains no copy of $\ell_1$ and $Y$ has the limited $p$-Schur property, then $K(X,Y)$ (the space of all compact operators from $X$ into $Y$) has the limited $p$-Schur property.