Chapter 11 Operator ideals

Abstract

This chapter explains the theory of operator ideals that was born in 1941 with the observation of Calkin. Schatten and von Neumann actually dealt with more general classes of operators; the so-called cross-spaces of linear transformations. They phrased their results in the language of direct (now tensor) products of the Banach spaces. To stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. The local theory of Banach spaces deals with those properties that can be expressed in terms of finite dimensional spaces. The most usual way is to consider the quantitative relations that are assumed to hold uniformly for any choice of n elements and/or functionals. Grothendieck's theorem has many applications in Banach space geometry. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal.