TL;DR
This paper characterizes when principal subideals of operator ideals are themselves ideals of B(H), extending previous work and exploring their lattice structure and relation to elementary operators.
Contribution
It provides necessary and sufficient conditions for principal subideals to be ideals of B(H), generalizing earlier results and analyzing their lattice structure.
Findings
Characterization of when principal subideals are ideals of B(H)
Complete description of all principal subideals
Analysis of the lattice structure of subideals
Abstract
A subideal is an ideal of an ideal of B(H) and a principal subideal is a principal ideal of an ideal of B(H). We determine necessary and sufficient conditions for a principal subideal to be an ideal of B(H). This generalizes to arbitrary ideals the 1983 work of Fong and Radjavi characterizing principal subideals of the ideal of compact operators that are also ideals of B(H). We then characterize all principal subideals. We also investigate the lattice structure of subideals as part of the general study of ideal lattices such as the often studied lattice structure of ideals of B(H). This study of subideals and the study of elementary operators with coefficient constraints are closely related.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Algebra and Logic · Holomorphic and Operator Theory