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Journal of Operator Theory

Volume 74, Issue 2, Fall 2015  pp. 371-389.

Nilpotent commutators with a masa

Authors:  Mitja Mastnak (1), Matjaz Omladic (2), and Heydar Radjavi (3)
Author institution: (1) Department of Mathematics, Saint Mary's University, Halifax, B3H 3C3, Canada
(2) Department of Mathematics, Institute of Mathematics, Physics and Mechanics, Ljubljana, SI-1000, Slovenia
(3) Department of Pure Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

Summary:  Let $\mathcal{H}$ be a complex Hilbert space, let $\mathcal{D}\subset \mathcal{B}(\mathcal{H})$ be a discrete masa (maximal abelian selfadjoint algebra) and let $\mathcal{A}$ be a linear subspace (or a singleton subset) of $\mathcal{B}(\mathcal{H})$ not necessarily having any nontrivial intersection with $\mathcal{D}$. Assume that the commutator $AD-DA$ is quasinilpotent for every $A\in\mathcal{A}$ and every $D\in\mathcal{D}$. We prove that $\mathcal{A}$ and $\mathcal{D}$ are simultaneously triangularizable. If $\mathcal{D}$ is a continuous masa, there exist compact operators satisfying this condition that fail to have a multiplicity-free triangularization together with $\mathcal{D}$. However, we prove an analogous result in the case where $\mathcal{A}$ is a finite-dimensional space of operators of finite rank.

Keywords:  reducibility, triangularizability, commutators, quasinilpotent operators, masa

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