Subideals of Operators – A Survey and Introduction to Subideal-Traces

Abstract

Operator ideals in B(H) are well understood and exploited but ideals inside them have only recently been studied starting with the 1983 seminal work of Fong and Radjavi and continuing with two recent articles by the authors of this survey. This article surveys this study embodied in these three articles. A subideal is a two-sided ideal of J (for specificity also called a J-ideal) for J an arbitrary ideal of B(H). In this terminology we alternatively call J a B(H)-ideal.This surveys [5], [13] and [14] in which we developed a complete characterization of all J-ideals generated by sets of cardinality strictly less than the cardinality of the continuum. So a central theme is the impact of generating sets for subideals on their algebraic structure. This characterization includes in particular finitely and countably generated J-ideals. It was obtained by first generalizing to arbitrary principal Jideals the 1983 work of Fong-Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. 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 K(H)-softness of B(H)-ideals introduced by Kaftal-Weiss and earlier exploited for Banach spaces by Mityagin and Pietsch. This study of subideals and the study of elementary operators with coefficient constraints are closely related. Here we also introduce and study a notion of subideal-traces where classical traces (unitarily invariant linear functionals) need not make sense for subideals that are not B(H)-ideals.

Exotic closed subideals of algebras of bounded operators

“…We will show that, surprisingly enough, the above property fails in general for closed subideals of L(X). We will adhere to the terminology suggested by Patnaik and Weiss [16], [17], and say that J is an I-subideal of L(X), if J ⊂ I, where I is an ideal of L(X) and J is an ideal of I. We will only be concerned with closed linear subideals, that is, J ⊂ I are closed linear subspaces of L(X), such that U S ∈ J and SU ∈ J whenever S ∈ J and U ∈ I (and similarly for I ⊂ L(X)).…”

“…[9,Theorem 1] or [16,Example 1.3]. References [16] and [17] also discuss ring-theoretic subideals of L(H), which will not play any role here.…”