Quasisimilar operators in the commutant of a cyclic subnormal operator

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.