Similarity to a contraction, for powerbounded operators with finite peripheral spectrum
Author:
Ralph deLaubenfels
Journal:
Trans. Amer. Math. Soc. 350 (1998), 31693191
MSC (1991):
Primary 47A05; Secondary 47A60, 47D03, 47A45, 47A10, 47A12
MathSciNet review:
1603894
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Abstract: Suppose is a powerbounded 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 halfplanes 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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 [Bod2]
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 R. deLaubenfels, Strongly continuous groups, similarity and numerical range on a Hilbert space, Taiwanese J. Math. 1 (1997), 127133. CMP 97:13
 [Fo]
 S. R. Foguel, A counterexample to a problem of Sz.Nagy, Proc. Amer. Math. Soc. 15 (1964), 788790. MR 29:2646
 [FrM]
 E. Franks and A. McIntosh, Discrete quadratic estimates and holomorphic functional calculi of operators in Banach spaces, Ulmer Seminare über Funktionalanalysis und Differentialgleichungen 1996, 155172.
 [G]
 J. A. Goldstein, ``Semigroups of Linear Operators and Applications,'' Oxford Univ. Press, New York, 1985. MR 87c:47056
 [H1]
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 [H2]
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 [M]
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 A. Pazy, ``Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, New York, 1983. MR 85g:47061
 [Pi]
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 [VC]
 J. A. van Casteren, ``Generators of Strongly Continuous Semigroups,'' Research Notes in Mathematics 115, Pitman, Boston, 1985. Zbl.576:47023
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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/S0002994798023034
PII:
S 00029947(98)023034
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
