Quasisimilar operators in the commutant of a cyclic subnormal operator

Raphael, Marc

doi:10.1090/S0002-9939-1985-0784176-4

Quasisimilar operators in the commutant of a cyclic subnormal operator
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by Marc Raphael

Proc. Amer. Math. Soc. 94 (1985), 265-268

DOI: https://doi.org/10.1090/S0002-9939-1985-0784176-4

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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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