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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A representation theorem for cyclic analytic two-isometries
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by Stefan Richter PDF
Trans. Amer. Math. Soc. 328 (1991), 325-349 Request permission

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.
References
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • 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