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)



The stability radius
of a quasi-Fredholm operator

Author: Pak Wai Poon
Journal: Proc. Amer. Math. Soc. 126 (1998), 1071-1080
MSC (1991): Primary 47A55, 47A10, 47A53
MathSciNet review: 1443849
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the technique used by Kordula and Müller to show that the stability radius of a quasi-Fredholm operator $T$ is the limit of $\gamma(T^n)^{1/n}$ as $n\rightarrow\infty$. If $0$ is an isolated point of the Apostol spectrum $\sigma _\gamma(T)$, then the above limit is non-zero if and only if $T$ is quasi-Fredholm.

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

  • 1. C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), 279-294. MR 87a:47003
  • 2. H. Bart and C. Lay, The stability radius of a bundle of closed linear operators, Studia Math. 66 (1980), 307-320. MR 82c:47014
  • 3. K.-H. Förster and M. A. Kaashoek, The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator, Proc. Amer. Math. Soc. 49 (1975), no. 1, 123-131. MR 51:8867
  • 4. S. Grabiner, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), no. 2, 317-337. MR 84a:47003
  • 5. V. Kordula and V. Müller, The distance from the Apostol spectrum, Proc. Amer. Math. Soc. 124 (1996), 3055-3061. MR 96m:47007
  • 6. J. Ph. Labrousse, Les opérateurs quasi Fredholm: Une généralisation des opérateurs semi Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), 161-258. MR 83c:47022
  • 7. J. Ph. Labrousse and M. Mbekhta, Résolvant généralisé et séparation des points singuliers quasi-Fredholm, Trans. Amer. Math. Soc. 333 (1992), no. 1, 299-313. MR 92k:47007
  • 8. E. Makai and J. Zemánek, The surjectivity radius, packing numbers and boundedness below of linear operators, Integral Equations Operator Theory 6 (1983), 372-384. MR 84m:47005
  • 9. M. Mbekhta and V. Müller, On the axiomatic theory of spectrum II, Studia Math. 119 (1996), 129-147. MR 97c:47005
  • 10. M. Mbekhta and A. Ouahab, Contribution à la théorie spectrale généralisée dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 833-836.
  • 11. V. Müller, On the regular spectrum, J. Operator Theory 31 (1994), 363-380.
  • 12. P. W. Poon, The Apostol representation of a linear operator, Preprint, Department of Mathematics, University of Melbourne.
  • 13. Ch. Schmoeger, The stability radius of an operator of Saphar type, Studia Math. 113 (1995), no. 2, 169-175. MR 96a:47019
  • 14. J. Zemánek, The stability radius of a semi-Fredholm operator, Integral Equations Operator Theory 8 (1985), 137-144. MR 86c:47014

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A55, 47A10, 47A53

Retrieve articles in all journals with MSC (1991): 47A55, 47A10, 47A53

Additional Information

Pak Wai Poon
Affiliation: Department of Mathematics, University of Melbourne, Victoria, 3052, Australia

Keywords: Stability radius, Apostol spectrum, semi-regular, quasi-Fredholm operators, ascent, descent
Received by editor(s): June 21, 1996
Received by editor(s) in revised form: September 23, 1996
Additional Notes: The results in this paper form a part of the author’s research for the degree of Ph.D. at the University of Melbourne, 1996, under the supervision of J. J. Koliha.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society