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

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

Published electronically:
April 27, 2000

MathSciNet review:
1664345

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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]**R. P. Boas :*Entire Functions*, Academic Press, New York, 1954.MR**16:914f****[2]**J. A. Deddens :*Another description of nest algebras in Hilbert spaces operators*, Lecture notes in Mathematics No. 693, (pp. 77-86), Springler-Verlag, Berlin, 1978.MR**80f:47033****[3]**B. Ja. Levin :*Distributions of Zeros of Entire Functions*, Amer. Math. Soc. Providence, 1964.MR**28:217****[4]**B. Ja. Levin :*Lectures on Entire Functions*, Translations of Mathematical Monographs, Vol. 150, American Mathematical Society, 1996.MR**97j:30001****[5]**G. Lumer and R. S. Phillips :*Dissipative operators in a Banach space*, Pacific J. Math. 11(1961), 679-698.MR**24:A2248****[6]**P. G. Roth :*Bounded orbits of conjugation, analytic theory*, Indiana Univ. Math. J., 32 (1983), 491-509.MR**85c:47039****[7]**W. Rudin :*Real & Complex Analysis*, Mc Graw-Hill, New York, 1966.MR**35:1420****[8]**J. G. Stampfli :*On a question of Deddens in Hilbert space operators*, Lecture Notes in Mathematics No. 693, (pp. 169-173), Springer-Verlag, Berlin, 1978.MR**80f:47034****[9]**J. G. Stampfli and J. P. Williams :*Growth conditions and the numerical range in a Banach algebra*, Tôhôku Math. J. 20(1968), 417-424.MR**39:4674****[10]**J. P. Williams :*On a boundedness condition for operators with singleton spectrum*, Proc. Amer. Math. Soc., 78(1980), 30-32.MR**81k:47008**

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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