Asplund operators and the Szlenk index

Abstract

This thesis constitutes an investigation into the Szlenk index of operators acting between Banach spaces, primarily in the context of operator ideal theory. Roughly speaking, the Szlenk index measures the complexity of certain sets in dual Banach spaces, and it has become a standard tool of the trade in Banach space theory since its introduction by W. Szlenk in 1968. The starting point of this thesis is a proof that there is a class of closed operator ideals naturally associated with the Szlenk index. We study their basic operator ideal properties and their relationship with the well-known operator ideals of separable range operators and Asplund operators. We provide a detailed analysis of the behaviour of the Szlenk index under taking direct sums of operators, ultimately applying the techniques we develop to this end in an investigation of factorisation properties of the closed operator ideals associated with the Szlenk index.