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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Quasisimilar operators in the commutant of a cyclic subnormal operator


Author: Marc Raphael
Journal: Proc. Amer. Math. Soc. 94 (1985), 265-268
MSC: Primary 47B20; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1985-0784176-4
MathSciNet review: 784176
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Abstract: Compactly supported positive regular Borel measures on the complex plane that share "certain" properties with normalized arclength measure on the boundary of the unit disk are called $ m$-measures. Let $ \mu $ be an $ m$-measure and let $ {S_\mu }$ be the cyclic subnormal operator of multiplication by $ z$ on the closure of the polynomials in $ {L^2}(\mu )$. Necessary and sufficient conditions for an operator in the commutant of $ {S_\mu }$ to be quasisimilar to $ {S_\mu }$ are investigated. In particular it is shown that if the Bergman shift and an operator in its commutant are quasisimilar, then they are unitarily equivalent.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0784176-4
Article copyright: © Copyright 1985 American Mathematical Society