Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Abstract | References | Similar Articles | Additional Information


For a bounded invertible operator $ A $ on a complex Banach space $ X,$ let $ B_A $ be the set of operators $ T $ in $ \mathcal{L} (X) $ for which $ \sup_{n \geq 0} \Vert A^n T A^{-n}\Vert < \infty.$ Suppose that $ Sp(A) = \{1\} $ and $ T $ is in $ B_A \cap B_{A^{-1}}. $ A bound is given on $ \Vert ATA^{-1} - T\Vert $ in terms of the spectral radius of the commutator. Replacing the condition $ T $ in $ B_{A^{-1}} $ by the weaker condition $\Vert A^{-n} T A^n\Vert = o(e^{\epsilon\sqrt{n}}), $ as $ n \to \infty$ for every $\epsilon>0$, an extension of the Deddens-Stampfli-Williams results on the commutant of $ A $ is given.

References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B10, 47B15

Retrieve articles in all journals with MSC (1991): 47B10, 47B15

Additional Information

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

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

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

American Mathematical Society