Some Classes of Operators

Abstract

The present chapter is devoted to the study of some of the most important classes of continuous linear operators between locally convex spaces, and in particular between Banach spaces. It is important to note that all these classes do have the ideal property, so that they furnish basic examples for the general theory of ideals to be considered in greater detail in Chapter 19 below. We start our investigations in 17.1 by presenting some fundamental results on compact operators. In particular, the relationship to Schwartz topologies is emphasized. 17.2 is devoted to weakly compact operators; especially, the factorization theorem for weakly compact operators with values in a Fréchet space is proved. In 17.3, we introduce nuclear operators and prove their basic properties. By examining the dual of an £-tensor product, we are led in 17.4 to the so-called integral operators of Grothendieck. Spaces of nuclear and integral operators between Banach spaces can be made into Banach spaces in a canonical fashion. The norm for integral operators relates in a simple way to the trace for finite operators. This is the starting point for our investigations in 17.5, culminating in Dean’s proof of one of the most important results in recent functional analysis: the principle of local reflexivity for Banach spaces. Finally, in 17.6, we discuss some particular cases, such as Grothendieck’s results on the representability of operators defined on an L₁ (μ)-space and with values in a Banach space, and on the equivalence of nuclear and integral operators.