Subideals of operators II

TL;DR

This paper characterizes subideals (J-ideals) within B(H)-ideals, extending previous work by identifying which countably generated J-ideals are B(H)-ideals and exploring properties like J-softness.

Contribution

It provides a complete characterization of countably generated J-ideals that are B(H)-ideals and introduces the concept of J-softness in this context.

Findings

Characterization of countably generated J-ideals that are B(H)-ideals.

Identification of J-ideals with generating sets smaller than the continuum as B(H)-ideals.

Introduction of J-softness as a key property in the structure of subideals.

Abstract

A subideal (also called a J-ideal) is an ideal of a B(H)-ideal J. This paper is the sequel to Subideals of operators where a complete characterization of principal and then finitely generated J-ideals were obtained by first generalizing the 1983 work of Fong and Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. Here we determine which countably generated J-ideals are B(H)-ideals, and in the absence of the continuum hypothesis which J-ideals with generating sets of cardinality less than the continuum are B(H)-ideals. These and some other results herein are based on the dimension of a related quotient space. We use this to characterize these J-ideals and settle additional questions about subideals. A key property in our investigation turned out to be J-softness of a B(H)-ideal I inside J, that is, IJ = I, a generalization of a recent notion of softness of B(H)-ideals introduced by Kaftal-Weiss and earlier exploited for Banach spaces by Mityagin and Pietsch.