Equicompact-Type Subsets of Operator Spaces

Abstract

A set \(H \subset K(X, Y)\) (the space of all compact operators between two Banach spaces X and Y) is said to be uniformly completely continuous (or sequentially weak-norm equicontionuous) if for each weakly null sequence \((x_n)\subset X\), the sequence \((T(x_n))\) converges uniformly on \(T\in H\). Also, \(H \subset K(X, Y)\) is called equicompact if every bounded sequence \((x_{n})\) in X has a subsequence \((x_{k(n)})_n\) such that \((Tx_{k(n)})_n\) is uniformly convergent for \(T\in H\). Using the Right topology, we study the concept of uniformly pseudo weakly compact (or sequentially Right-norm equicontionuous) and also uniformly limited completely continuous subsets of some operator spaces. In particular, in terms of completely continuous operators into \(c_0\), we give an operator characterization of those subsets of L(X, Y) whose uniformly pseudo weakly compact (or uniformly limited completely continuous) sets are uniformly completely continuous and also those subsets of L(X, Y) whose uniformly completely continuous sets are equicompact.