[phi]-Summing operators in Banach spaces

Abstract

Let [phi]: [0, [infinity]) --> [0, [infinity]) be a continuous subadditive strictly increasing function and [phi](0) = 0. Let E and F be Banach spaces. A bounded linear operator A: E --> F will be called [phi]-summing operator if there exists [lambda] > 0 such that [summation operator]i = 1n [phi] ||Axi|| [les][lambda] sup||x*|| [les] 1 [epsilon]i = 1n [phi] |xi, x*>|, for all sequences {x1, ..., xn [subset of or equal to] E}. We set [Pi][phi](E, F) to denote the space of all [phi]-summing operators from E to F. We study the basic properties of the space [Pi][phi](E, F). In particular, we prove that [Pi][phi](H, H) = [Pi]p(H, H) for 0 [les] p H is a Banach space with the metric approximation property.