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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Similarity to a contraction, for power-bounded operators with finite peripheral spectrum
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by Ralph deLaubenfels PDF
Trans. Amer. Math. Soc. 350 (1998), 3169-3191 Request permission

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 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.
  • © Copyright 1998 American Mathematical Society
  • 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