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

Full-text PDF Free Access

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.

**[Be]**Bernard Beauzamy,*Introduction to operator theory and invariant subspaces*, North-Holland Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR**967989****[Bod1]**K. Boyadzhiev and R. deLaubenfels,*Semigroups and resolvents of bounded variation, imaginary powers and 𝐻^{∞} functional calculus*, Semigroup Forum**45**(1992), no. 3, 372–384. MR**1179859**, https://doi.org/10.1007/BF03025777**[Bod2]**Khristo Boyadzhiev and Ralph deLaubenfels,*Spectral theorem for unbounded strongly continuous groups on a Hilbert space*, Proc. Amer. Math. Soc.**120**(1994), no. 1, 127–136. MR**1186983**, https://doi.org/10.1090/S0002-9939-1994-1186983-0**[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**0165362**, https://doi.org/10.1090/S0002-9939-1964-0165362-X**[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]**Jerome A. Goldstein,*Semigroups of linear operators and applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR**790497****[H1]**P. R. Halmos,*On Foguel’s answer to Nagy’s question*, Proc. Amer. Math. Soc.**15**(1964), 791–793. MR**0165363**, https://doi.org/10.1090/S0002-9939-1964-0165363-1**[H2]**P. R. Halmos,*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887–933. MR**0270173**, https://doi.org/10.1090/S0002-9904-1970-12502-2**[K]**Tosio Kato,*Note on fractional powers of linear operators*, Proc. Japan Acad.**36**(1960), 94–96. MR**0121666****[LM]**C. Le Merdy,*The similarity problem for bounded analytic semigroups on Hilbert space,*Semigroup Forum, to appear.**[M]**Alan McIntosh,*Operators which have an 𝐻_{∞} functional calculus*, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR**912940****[Pau1]**Vern I. Paulsen,*Every completely polynomially bounded operator is similar to a contraction*, J. Funct. Anal.**55**(1984), no. 1, 1–17. MR**733029**, https://doi.org/10.1016/0022-1236(84)90014-4**[Pau2]**Vern I. Paulsen,*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472****[Paz]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[Pi]**Gilles Pisier,*A polynomially bounded operator on Hilbert space which is not similar to a contraction*, J. Amer. Math. Soc.**10**(1997), no. 2, 351–369. MR**1415321**, https://doi.org/10.1090/S0894-0347-97-00227-0**[R]**Gian-Carlo Rota,*On models for linear operators*, Comm. Pure Appl. Math.**13**(1960), 469–472. MR**0112040**, https://doi.org/10.1002/cpa.3160130309**[VC]**J. A. van Casteren, ``Generators of Strongly Continuous Semigroups,'' Research Notes in Mathematics 115, Pitman, Boston, 1985. Zbl.576:47023

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
47A05,
47A60,
47D03,
47A45,
47A10,
47A12

Retrieve articles in all journals with MSC (1991): 47A05, 47A60, 47D03, 47A45, 47A10, 47A12

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