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

 

Similarity to a contraction, for power-bounded
operators with finite peripheral spectrum


Author: Ralph deLaubenfels
Journal: Trans. Amer. Math. Soc. 350 (1998), 3169-3191
MSC (1991): Primary 47A05; Secondary 47A60, 47D03, 47A45, 47A10, 47A12
MathSciNet review: 1603894
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Abstract: Suppose $T$ is a power-bounded linear opertor on a Hilbert space with finite peripheral spectrum (spectrum on the unit circle). Several sufficient conditions are given for $T$ to be similar to a contraction. A natural growth condition on the resolvent in half-planes tangent to the unit circle at the peripheral spectrum is shown to be equivalent to $T$ having an $H^\infty(\mathcal P)\cap C(\overline{\mathcal P})$ functional calculus, for some open polygon $\mathcal P$ contained in the unit disc, which, in turn, is equivalent to $T$ being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of $T$ be square summable also implies that $T$ is similar to a contraction.


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Additional Information

Ralph deLaubenfels
Affiliation: Scientia Research Institute, P. O. Box 988, Athens, Ohio 45701
Email: 72260.2403@compuserve.com

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02303-4
PII: S 0002-9947(98)02303-4
Received by editor(s): August 28, 1996
Additional Notes: I am indebted to Vũ Quôc Phóng and Christian Le Merdy for invaluable discussions; in particular, to Christian Le Merdy for sending me a preprint of [LM] and pointing out Lemma 1.6, and to Vũ Quôc Phóng for Lemma 3.13.
Article copyright: © Copyright 1998 American Mathematical Society