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

DOI:
https://doi.org/10.1090/S0002-9947-98-02303-4

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

Received by editor(s):
August 28, 1996

Additional Notes:
I am indebted to VĆ© QuĂŽc PhĂłng 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 VĆ© QuĂŽc PhĂłng for Lemma 3.13.

Article copyright:
© Copyright 1998
American Mathematical Society