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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 w-measures. Let u be an m-measure and let Ss be the cyclic subnormal operator of multiplication by ; on the closure of the polynomials in L2(ix). Necessary and sufficient conditions for an operator in the commutant of S^ to be quasisimilar to S^ 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. 1. Introduction. The canonical model for a cyclic subnormal operator is the operator S of multiplication by z on P2(jtt), the closure of the analytic polynomials in L2(p). Here p is any compactly supported positive regular Borel measure on the complex plane C. From now on such a jti will be referred to as a measure. The problem of determining quasisimilarity between cyclic subnormal operators is, in general, difficult. However, W. S. Clary (1) has given necessary and sufficient conditions for a subnormal operator to be quasisimilar to the unilateral shift, and J. B. Conway (4) has shown that if the unilateral shift and an operator in its commutant are quasisimilar, then they are unitarily equivalent. This paper investigates the equivalence relation of quasisimilarity between S and QiSp) when p is a meas"ure with many of the same properties as normalized arclength measure m on the boundary of the open unit disk D, tj> g P2(p) n L°°(ju), and 5M is the operator of multiplication by z on P2(p). The reader should bear in mind Yoshino's Theorem (9) that states if ju is a measure, then the commutant of S^ is

Quasisimilar operators in the commutant of a cyclic subnormal operator