$B(H)$-Commutators: A Historical Survey II and recent advances on commutators of compact operators
Daniel Beltita, Sasmita Patnaik, and Gary Weiss

TL;DR
This paper surveys historical and recent advances on the single commutator problem for compact operators, including positive operators, staircase forms, and Lie algebra frameworks, resolving longstanding questions.
Contribution
It provides new affirmative constructions of positive operators as commutators of compact operators and connects these results to staircase matrix forms and Lie algebra structures.
Findings
Constructed compact operators with rank one projections as commutators.
Identified classes of positive operators that are commutators of compact operators.
Extended commutator problem solutions to complex symplectic and semisimple Lie algebras.
Abstract
A sequel to \cite{gW05}, we address again the single commutator problem \cite{PT71} of Pearcy and Topping: Is every compact operator a single commutator of compact operators? by focusing on a 35 year old test question for this posed in 1976 by the last named author and others: Are there any strictly positive operators that are single commutators of compact operators? The latter we settle here affirmatively with a modest modification of Anderson's fundamental construction \cite{jA77} constructing compact operators whose commutator is a rank one projection. Moreover we provide here a rich class of such strictly positive operators that are commutators of compact operators and pose a question for the rest. We explain also how these methods are related to the study of staircase matrix forms, their equivalent block tri-diagonal forms, and commutator problems. In particular, we present the…
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Taxonomy
TopicsHolomorphic and Operator Theory · Advanced Operator Algebra Research · Advanced Harmonic Analysis Research
