2013/03/20 by Daniel Beltiţă, Daniel Beltita, Sasmita Patnaik +4
Mathematics · #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #Holomorphic and Operator Theory #math.FA #math.OA #math.RT #msc:17B22 #msc:17B65 #msc:47-02 #msc:47B10 #msc:47B47 #msc:47L20 #msc:47L30
paper · pdf · doi:10.48550/arxiv.1303.4844
20 pages
arxiv created 2013/03/20 · arxiv updated 2013/03/21
A sequel to \citegW05, we address again the single commutator problem \citePT71 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 \citejA77 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 original test question and solution that led to the negative solution of the Pearcy-Topping question on whether or not every trace class trace zero operator was a commutator (or linear combination of commutators) of Hilbert-Schmidt operators. And we show how this evolved from staircase form considerations along with a Larry Brown result on trace connections to ideals \citelB94 which itself is at the core of \cite[Section 7]DFWW. The omission in \citegW05 of this important 35 year old test question was inadvertent and we correct that in this paper. This sequel starts where [ibid] left off but can be read independently of [ibid]. The present paper also has a section on self-commutator equations [X^*,X]=A within the framework of some classical operator Lie algebras. That problem was solved by Fan and Fong (1980) for the full algebra of compact operators, and we solve it here for the complex symplectic Lie algebra of compact operators and for complex semisimple Lie algebras.