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 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.

**[Be]**B. Beauzamy, ``Introduction to Operator Theory and Invariant Subspaces,'' North-Holland, Amsterdam, 1988. MR**90d:47001****[Bod1]**K. Boyadzhiev and R. deLaubenfels,*Semigroups and resolvents of bounded variation, imaginary powers and functional calculi*, Semigroup Forum 45 (1992), 372-384. MR**93i:47018****[Bod2]**K. Boyadzhiev and R. deLaubenfels,*Spectral theorem for unbounded strongly continuous groups on a Hilbert space,*Proc. Amer. Math. Soc. 120 (1994), 127-136. MR**94c:47058****[d]**R. deLaubenfels,*Strongly continuous groups, similarity and numerical range on a Hilbert space*, Taiwanese J. Math. 1 (1997), 127-133. CMP**97:13****[Fo]**S. R. Foguel,*A counterexample to a problem of Sz.-Nagy,*Proc. Amer. Math. Soc. 15 (1964), 788-790. MR**29:2646****[Fr-M]**E. Franks and A. McIntosh,*Discrete quadratic estimates and holomorphic functional calculi of operators in Banach spaces,*Ulmer Seminare über Funktionalanalysis und Differentialgleichungen 1996, 155-172.**[G]**J. A. Goldstein, ``Semigroups of Linear Operators and Applications,'' Oxford Univ. Press, New York, 1985. MR**87c:47056****[H1]**P. R. Halmos,*On Foguel's answer to Nagy's question,*Proc. Amer. Math. Soc. 15 (1964), 791-793. MR**29:2647****[H2]**P. R. Halmos,*Ten problems in Hilbert space,*Bull. Amer. Math. Soc. 76 (1970), 887-933. MR**42:5066****[K]**T. Kato,*Note on fractional powers of linear operators,*Proc. Japan Acad. 36 (1960), 94-96. MR**22:12400****[LM]**C. Le Merdy,*The similarity problem for bounded analytic semigroups on Hilbert space,*Semigroup Forum, to appear.**[M]**A. McIntosh,*Operators which have an functional calculus,*Miniconference on Operator Theory and Partial Differential Equations, Proc. Center Math. Anal., ANU, vol. 14, Australian Nat. Univ., Canberra, 1986, pp. 210-231. MR**88k:47019****[Pau1]**V. I. Paulsen,*Every completely polynomially bounded operator is similar to a contraction,*J. Funct. Anal. 55 (1984), 1-17. MR**86c:47021****[Pau2]**V. I. Paulsen, ``Completely Bounded Maps and Dilations,'' Pitman Research Notes in Math. 146, Longman Sci. Tech., Harlow, and Wiley, New York, 1986. MR**88h:46111****[Paz]**A. Pazy, ``Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer, New York, 1983. MR**85g:47061****[Pi]**G. Pisier,*A polynomially bounded operator on Hilbert space which is not similar to a contraction,*J. Amer. Math. Soc. 10 (1997), 351-369. MR**97f:47002****[R]**G. C. Rota,*On models for linear operators,*Comm. Pure Appl. Math. 13 (1960), 469-472. MR**22:2898****[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:
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