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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 Kmid/ (p,r)(X, Y ). We prove that the ideal (Kmid/(p,r)(X,Y),κmid/ (p,r)(·)) 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 Kmid/ (p,r)(X, Y ) consists of (p, r)-limited sets.
