The Gelfand--Phillips property in closed subspaces of some operator spaces

Abstract

By introducing the concept of limited completely continuous operators between two arbitrary Banach spaces $X$ and $Y$, we give some properties of this concept related to some well known classes of operators and specially, related to the Gelfand-Phillips property of the space $X$ or $Y$. Then some necessary and sufficient conditions for the Gelfand--Phillips property of closed subspace $M$ of some operator spaces, with respect to limited complete continuity of some operators on $M$, so-called, evaluation operators, are verified.