Let V = K0 © Vx be an isometry, where V0 is unitary and K, is a unilateral shift of finite multiplicity n. Let S = S0 © 5, be a subnormal operator where S0 © 5, is the normal decomposition of S into a normal operator S0 and a completely nonnormal operator S,. It is shown that 5 is quasisimilar to V if and only if S0 is unitarily equivalent to V0 and 5] is quasisimilar to V¡. To prove this, a standard representation is developed for n-cyclic subnormal operators. Using this representation, the

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... ss of subnormal operators which are quasisimilar to K, is completely characterized. 0. Introduction. The purpose of this paper is to study quasisimilarity within the class of subnormal operators. Two Hubert space operators A and B are quasisimilar if there exist operators X and Y which are one-to-one, have dense range, and satisfy XA = BX and A Y = YB. Quasisimilarity was introduced by Sz.-Nagy and Foia §, who gave a simple characterization of the class of operators which are quasisimilar to a unitary operator [13] . They also showed that if A is quasisimilar to a unitary operator W, then there is a one-to-one correspondence between the hyperinvariant subspaces of A and the hyperinvariant subspaces of W. In general, it is known that if A and B are quasisimilar and A has a hyperinvariant subspace, then so does B [7], An operator S is subnormal if S has an extension T to a larger Hubert space on which T is normal. Let ju be a positive, finite Borel measure with compact support in C and let i/M be the operation of multiplication by z on H2(n), the closure in L2(u) of the polynomials in z. Up to unitary equivalence, the operator U^ is the most general cyclic subnormal operator. If m is Lebesgue measure on 9" = {z: \z\ = 1}, then H2 = H2(m) is the classical Hardy space and Um is the unilateral shift. A measure u is of type S if (1) /i is carried by {z: \z\ < 1}, (2) the restriction of /x to 5" is absolutely continuous,