Operators with bounded conjugation orbits

Authors:
D. Drissi and M. Mbekhta

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2687-2691

MSC (1991):
Primary 47B10, 47B15

Published electronically:
April 27, 2000

MathSciNet review:
1664345

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every , an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

**[1]**Ralph Philip Boas Jr.,*Entire functions*, Academic Press Inc., New York, 1954. MR**0068627****[2]**James A. Deddens,*Another description of nest algebras*, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 77–86. MR**526534****[3]**B. Ja. Levin,*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975****[4]**B. Ya. Levin,*Lectures on entire functions*, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR**1400006****[5]**G. Lumer and R. S. Phillips,*Dissipative operators in a Banach space*, Pacific J. Math.**11**(1961), 679–698. MR**0132403****[6]**Paul G. Roth,*Bounded orbits of conjugation, analytic theory*, Indiana Univ. Math. J.**32**(1983), no. 4, 491–509. MR**703280**, 10.1512/iumj.1983.32.32035**[7]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528****[8]**Joseph G. Stampfli,*On a question of Deddens*, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 169–173. MR**526546****[9]**J. G. Stampfli and J. P. Williams,*Growth conditions and the numerical range in a Banach algebra*, Tôhoku Math. J. (2)**20**(1968), 417–424. MR**0243352****[10]**J. P. Williams,*On a boundedness condition for operators with a singleton spectrum*, Proc. Amer. Math. Soc.**78**(1980), no. 1, 30–32. MR**548078**, 10.1090/S0002-9939-1980-0548078-6

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Additional Information

**D. Drissi**

Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait

Email:
drissi@math-1.sci.kuniv.edu.kw

**M. Mbekhta**

Affiliation:
URA 751 au CNRS & UFR de Mathematiques, Université de Lille I, F-59655, Villeneuve d’asq, France;
Université de Galatasaray, Ciragan Cad no. 102, Ortakoy 80840, Istanbul, Turquie

Email:
Mostafa.Mbekhta@univ-lille1.fr

DOI:
https://doi.org/10.1090/S0002-9939-00-05338-7

Keywords:
Bounded conjugation orbit,
spectrum,
spectral radius

Received by editor(s):
June 23, 1998

Received by editor(s) in revised form:
October 27, 1998

Published electronically:
April 27, 2000

Additional Notes:
The first author acknowledges support from Kuwait University

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society