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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
DOI: https://doi.org/10.1090/S0002-9947-1991-1013337-1
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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Article copyright: © Copyright 1991 American Mathematical Society