Strongly limited (Dunford–Pettis) completely continuous subspaces of operator ideals

Abstract

By introducing the concepts of strongly limited completely continuous and strongly Dunford–Pettis completely continuous subspaces of operator ideals, it will be given some characterizations of these concepts in terms of lcc ness and DPcc ness of all their evaluation operators related to that subspace. In particular, when $$X^*$$ or Y has the Gelfand–Phillips (GP) property, we give a characterization of GP property of a closed subspace $${\mathcal {M}}$$ of compact operators K(X, Y) in terms of strong limited complete continuity of $${\mathcal {M}}$$ . Also it is shown that, each operator ideal $${\mathcal {U}}(X, Y )$$ is strongly limited completely continuous, iff, $$X^*$$ and Y have the GP property.