Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 97-124.

Around the closures of the set of commutators and the set of differences of idempotent elements of $\mathcal{B}(\mathcal{H})$

Authors: Laurent W. Marcoux (1), Heydar Radjavi (2), Yuanhang~Zhang (3)
Author institution: (1) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
(2) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
(3) School of Mathematics, Jilin Univ., Changchun, 130012, P.R. China

Summary: We describe the norm-closures of the set ${\mathfrak C}_{{\mathfrak E}}$ of commutators of idempotent operators and the set ${\mathfrak E} - {\mathfrak E}$ of differences of idempotent operators acting on a finite-dimensional complex Hilbert space, as well as characterise the intersection of the closures of these sets with the set ${\mathcal{K} ( \mathcal{H})}$ of compact operators acting on an infinite-dimensional complex separable Hilbert space ${\mathcal H}$. Finally, we characterise the closures of the set ${\mathfrak C}_{{\mathfrak P}}$ of commutators of orthogonal projections and the set ${\mathfrak P} - {\mathfrak P}$ of differences of orthogonal projections acting on a complex separable Hilbert space.

DOI: http://dx.doi.org/10.7900/jot.2022feb07.2396
Keywords: commutators, differences, idempotents, projections, closures

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