Compactness property associated with the quasi-normed integral operator ideals

Abstract

In this thesis, we conduct a study on the (p, r)-compactness and mid (p, r)-compactness of subsets in Banach spaces for 1 ≤ p ≤ ∞, and 1 ≤ r ≤ p∗, where p∗ is the conjugate index of p. We begin by introducing and studying a compactness property which a Banach space may or may not have. This compactness property will be denoted by C_p^r and it is the class of all Banach spaces X such that X belongs to C_p^r if for every bounded subset A of X, A is relatively (p, r)-compact if, and only if, U_A^∗ belongs to the injective hull of the (p, r∗, 1)-integral operators where U_A^∗ is the adjoint of the operator U_A : ℓ_1(A) → X. Our main interest is to investigate the relationship between the (p, r)-compactness of sets and the C_p^r Property of Banach spaces. Moreover, we will also prove a characterization that a Banach space Y has the C_p^r Property precisely when the (p, r)-compact operators from X into Y equals the surjective hull of the dual of the (p, r∗, 1)-integral operators from X into Y for every Banach space X. Other results with regard to the C_p^r Property of Banach spaces will also be proved. We also 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 K^mid_(p,r)(X, Y ). We prove that the ideal (K^mid_(p,r)(X, Y ), κ^mid_(p,r)(·)) is a quasi-Banach operator ideal. Finally, we introduce and study the (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 K^mid_(p,r)(X, Y ) consists of (p, r)-limited sets. Other results with regard to this ideal (K^mid_(p,r)(X, Y ), κ^mid_(p,r)(·)) and the (p, r)-limited sets will also be proved.