Journal of Operator Theory

Volume 82, Issue 1, Summer 2019 pp. 79-113.

Geometry of joint spectra and decomposable operator tuples

Authors: M.I. Stessin (1), A.B. Tchernev (2)
Author institution:(1) Department of Mathematics and Statistics, University at Albany, SUNY, Albany, NY 12222, U.S.A.
(2) Department of Mathematics and Statistics, University at Albany, SUNY, Albany, NY 12222, U.S.A.

Summary: Joint spectra of tuples of operators are subsets in complex projective space. We investigate the relationship between the geometry of the spectrum and the properties of the operators in the tuple when these operators are self-adjoint. In the case when the spectrum contains an algebraic hypersurface passing through an isolated spectral point of one of the operators we give necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. We also address spectral continuity and obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines.

DOI: http://dx.doi.org/10.7900/jot.2018feb21.2227
Keywords: joint spectrum, algebraic curves

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