The ideal of weakly compactly generated operators acting on a Banach space

TL;DR

This paper introduces and thoroughly analyzes the class of weakly compactly generated (WCG) operators on Banach spaces, establishing their properties, relations to classical ideals, and applications to the structure of operator algebras.

Contribution

It defines WCG operators, proves they form a closed surjective operator ideal, and explores their role in the structure of operator algebras, including applications to the long James space.

Findings

WCG operators form a closed surjective operator ideal.

The ideal of WCG operators is the unique maximal ideal in certain contexts.

The $K_0$-group of $ ext{B}( ext{J}_p( ext{ω}_1))$ is isomorphic to the integers.

Abstract

We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\mathcal{J}_p(\omega_1)$, we show how properties of the ideal of $\mathsf{WCG}$ operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the $K_0$-group of $\mathscr{B}(\mathcal{J}_p(\omega_1))$ as the additive group of integers.