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)



On the poles of the resolvent in Calkin algebra

Author: O. Bel Hadj Fredj
Journal: Proc. Amer. Math. Soc. 135 (2007), 2229-2234
MSC (2000): Primary 47L10
Published electronically: March 2, 2007
MathSciNet review: 2299500
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space $ H$. We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator $ T$ in the Calkin algebra, then there exists a compact operator $ K$ for which zero is a pole of $ T+K$ if and only if the index of $ T-\lambda$ is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.

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

  • 1. C. APOSTOL, The reduced minimum modulus, Michigan Math. J. 32 1985, 276-294. MR 0803833 (87a:47003)
  • 2. B. AUPETIT, E. MAKAI JR., M. MBEKHTA, J. ZEMÁNEK, The connected components of the idempotents in the Calkin algebra, and their liftings, Operator theory and Banach algebras (Rabat, 1999), Theta, Bucharest, 2003, 23-30. MR 2006311 (2004g:46062)
  • 3. S. K. BERBERIAN, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 1970, 529-544. MR 0279623 (43:5344)
  • 4. F. F. BONSALL, J. DUNCAN, Complete Normed Algebras, Ergebnisse der Mathematik und ihrer Grenzgebeite, Band 80, Springer Verlag, New York-Heidelberg, 1973. MR 0423029 (54:11013)
  • 5. M. BURGOS, A. KAIDI, M. MBEKHTA, M. OUDGHIRI, On the descent spectrum, accepted for publication in J. Oper. Theory.
  • 6. M. BURGOS, O. BEL HADJ FREDJ, M. OUDGHIRI, Ascent and essential ascent spectrum, Preprint (2005).
  • 7. S.R. CARADUS, W.E. PFAFFENBERGER, Y. BERTRAM , Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, New York, 1974. MR 0415345 (54:3434)
  • 8. P. DE LA HARPE, Initiation à l'algèbre de Calkin (French), Algèbres d'oprateurs (Sém., Les Plans-sur-Bex, 1978), pp. 180-219, Lecture Notes in Math., 725, Springer, Berlin, 1979. MR 0548115 (81c:47045)
  • 9. R. G. DOUGLAS, Banach Algebra Techniques in Operator Theory, Academic Press 1972. MR 0361893 (50:14335)
  • 10. S. GRABINER, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982) 317-337. MR 0651274 (84a:47003)
  • 11. S. GRABINER AND J. ZEMÁNEK, Ascent, descent, and ergodic properties of linear operators, J. Oper. Theory 48 (2002) 69-81. MR 1926044 (2003g:47024)
  • 12. D. C. LAY, Spectral analysis using ascent, descent, nullity and defect , Math. Ann. 184 (1970) 197-214. MR 0259644 (41:4279)
  • 13. V. MULLER, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, 139. Birkhäuser Verlag, Basel, 2003. MR 1975356 (2004c:47008)
  • 14. C. L. OLSEN, A structure theorem for polynomially compact operators. Amer. J. Math. 93 (1971), 686-698. MR 0405152 (53:8947)
  • 15. B. N. SADOVSKII, Limit-compact and condensing operators, (Russian) Uspehi Mat. Nauk 163 (1972), 81-146 (Russian), English transl. Russian Math. Surveys 27 (1972), 85-155. MR 0428132 (55:1161)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47L10

Retrieve articles in all journals with MSC (2000): 47L10

Additional Information

O. Bel Hadj Fredj
Affiliation: Université Lille 1, UFR de Mathématiques, UMR-CNRS 8524, 59655 Villeneuve d’Ascq, France

Keywords: Poles, Calkin algebra, essential ascent and essential descent
Received by editor(s): February 6, 2006
Received by editor(s) in revised form: March 28, 2006
Published electronically: March 2, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society