On mid (p, r)-compact operators

Abstract

Let 1 ≤ p ≤ ∞ and 1 ≤ r ≤ p* where p* is the conjugate index of p. We introduce and study mid (p, r)-compact sets and operators. We begin by introducing and defining the mid (p, r)-compact subsets of a Banach space X and the mid (p, r)-compact operators between Banach spaces X and Y. The set of mid (p, r)-compact operators between Banach spaces X and Y is denoted by . We prove that the ideal is a quasi-Banach operator ideal. We also introduce and study (p, r)-limited subsets in Banach spaces. We prove that every mid (p, r)-compact subset of X is (p, r)-limited and that the set consists of (p, r)-limited sets.