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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A representation theorem for cyclic analytic two-isometries


Author: Stefan Richter
Journal: Trans. Amer. Math. Soc. 328 (1991), 325-349
MSC: Primary 47B38; Secondary 31C25, 47A15
MathSciNet review: 1013337
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Abstract: A bounded linear operator $ T$ on a complex separable Hilbert space $ \mathcal{H}$ is called a $ 2$-isometry if $ {T^{\ast 2}}{T^2} - 2{T^{ \ast }}T + I = 0$. We say that $ T$ is analytic if $ { \cap _{n> 0}}\,{T^n}\,\mathcal{H}= (0)$. In this paper we show that every cyclic analytic $ 2$-isometry can be represented as multiplication by $ z$ on a Dirichlet-type space $ D(\mu)$. Here $ \mu $ denotes a finite positive Borel measure on the unit circle. For two measures $ \mu $ and $ \nu $ the $ 2$-isometries obtained as multiplication by $ z$ on $ D(\mu)$ and $ D(\nu)$ are unitarily equivalent if and only if $ \mu = \nu $. We also investigate similarity and quasisimilarity of these $ 2$-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1013337-1
PII: S 0002-9947(1991)1013337-1
Article copyright: © Copyright 1991 American Mathematical Society