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 m -measures. Let μ \mu be an m m -measure and let S μ {S_\mu } be the cyclic subnormal operator of multiplication by z z on the closure of the polynomials in L 2 ( μ ) {L^2}(\mu ) . Necessary and sufficient conditions for an operator in the commutant of S μ {S_\mu } to be quasisimilar to S μ {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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