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.
DOI: http://dx.doi.org/10.7900/jot.2014jul02.2060
Keywords: reducibility, triangularizability, commutators,
quasinilpotent operators, masa
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