On quasisimilarity for subnormal operators

Abstract

In this paper it is shown that if two subnormal operators on separable Hilbert spaces are quasisimilar, then the weak* (or, ultraweakly)closed algebras they generate are isomorphic, and this isomorphism has additional properties. The pure and normal parts of quasisimilar subnormal operators are also investigated, and it is shown that the normal parts must be unitarily equivalent. It is also proved that the pure parts of quasisimilar cyclic subnor- maloperators must be quasisimilar. These results are then applied to character- ize the normal operators all of whose simple parts are quasisimilar. Another application is made to obtain part of a result of W. S. Clary characterizing those subnormal operators that are quasisimilar to the unilateral shift of multi- plicity one.