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 | References | Similar Articles | Additional Information

Abstract: Suppose 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 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 having an functional calculus, for some open polygon contained in the unit disc, which, in turn, is equivalent to being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of be square summable also implies that 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:
https://doi.org/10.1090/S0002-9947-98-02303-4

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