Surjective ?-compactness and generalized Kolmogorov numbers

Abstract

B. Carl and I. Stephani [2,3] have introduced a refined notion of $?$-compactness where $?$ is a given operator ideal; in a more recent paper, I. Stephani [6] has introduced the concept of in-jectively $ ?$-compact operators in view of which the Carl-Stephani notion of $?$-compactness will be more appropriately one of surjective $?$- compactness.Carl and Stephani also defined generalized outer entropy numbers in terms of which (surjective) $?$-compactness is characterized while Stephani characterizes injective $?$-compactness of operators in terms of their generalized inner entropy numbers and generalized Gelfand numbers.In this paper we give an appropriate generalization of Kolmogorov numbers of sets and operators with reference to a fixed quasi-normed ideal and characterize surjective $?$-compact-ness in terms of these numbers. Also, for a set H of operators we define their measures of equi-$?$-variation and apply these ideas in studying $?$-compact sets of $?$-compact operators and present results analogous to those of K. Vala [7].