Absolutely summing operators on separable Lindenstrauss spaces as tree spaces and the bounded approximation property

Abstract

Let X be a Banach space and let Y be a separable Lindenstrauss space. We describe the Banach space P(Y,X) of absolutely summing operators as a general '1-tree space. We also characterize the bounded approximation property and its weak version for X in terms of the space of integral operators I(X,Z ) and the space of nuclear operators N (X,Z ), respectively, where Z is a Lindenstrauss space, whose dual Z fails to have the Radon-Nikodym property.