Subnormal operators quasisimilar to an isometry

Hastings, William W.

doi:10.1090/S0002-9947-1979-0546912-3

Subnormal operators quasisimilar to an isometry
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by William W. Hastings

Trans. Amer. Math. Soc. 256 (1979), 145-161

DOI: https://doi.org/10.1090/S0002-9947-1979-0546912-3

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Abstract:

Let $V = {V_0} \oplus {V_1}$ be an isometry, where ${V_0}$ is unitary and ${V_1}$ is a unilateral shift of finite multiplicity n. Let $S = {S_0} \oplus {S_1}$ be a subnormal operator where ${S_0} \oplus {S_1}$ is the normal decomposition of S into a normal operator ${S_0}$ and a completely nonnormal operator ${S_1}$. It is shown that S is quasisimilar to V if and only if ${S_0}$ is unitarily equivalent to ${V_0}$ and ${S_1}$ is quasisimilar to ${V_1}$. To prove this, a standard representation is developed for n-cyclic subnormal operators. Using this representation, the class of subnormal operators which are quasisimilar to ${V_1}$ is completely characterized.

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