Compact operators which are defined by l _p-spaces

Abstract

Let 1 ≤ p < ∞ and 1 ≤ r ≤ p*, where p* is the conjugate index of p. We say that a linear operator T from a Banach space X to a Banach space Y is (p, r)-compact if the image of the unit ball T(B _X ) is contained in (where (a _n ) ∈ B_c0 if r = ∞) for some p-summable sequence (y _n ) ∈ l _p (Y). This encompasses the notion of p-compact operators which are precisely the (p, p*)-compact operators. We describe the quasi-Banach operator ideal structure of the class of (p, r)-compact operators. Our approach is more direct and easier than those that have been developed so far for the study of the class of p-compact operators.