Rendiconti del Circolo Matematico di Palermo

, Volume 51, Issue 3, pp 495–502

On operators with closed analytic core

  • T. Len Miller
  • Vivien G. Miller
  • Michael M. Neumann


As shown by Mbekhta [9] and [10], the analytic core and the quasi-nilpotent part of an operator play a significant role in the local spectral and Fredholm theory of operators on Banach spaces. It is a basic fact that the analytic core is closed whenever 0 is an isolated point of the spectrum. In this note, we explore the extent to which the converse is true, based on the concept of support points. Our results are exemplified in the case of decomposable operators, Riesz operators, convolution operators, and semi-shifts.

2000 Mathematics Subject Classification

47A11 47A10 47B40 47B06 

Key words and phrases

Local spectral theory decomposable operators Riesz operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aiena P., Colasante M. L., González M.,Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc.,130 (2002), 2701–2710.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Aiena P., Miller T. L., Neumann M. M.,On a localized single-valued extension property, to appear in Proc. Royal Irish Acad.Google Scholar
  3. [3]
    Caradus S. R., Pfaffenberger W. E., Yood B.,Calkin algebras and algebras of operators on Banach spaces, Marcel Dekker, New York, 1974.MATHGoogle Scholar
  4. [4]
    ColojoaraĂ I., Foiaş C.,Theory of generalized spectral operators, Gordon and Breach, New York, 1968.Google Scholar
  5. [5]
    Dowson H. R.,Spectral theory of linear operators, Academic Press, London, 1978.MATHGoogle Scholar
  6. [6]
    Eschmeier J.,Operator decomposability and weakly continuous representations of locally compact abelian groups, J. Operator Theory,7 (1982), 201–208.MATHMathSciNetGoogle Scholar
  7. [7]
    Laursen K. B., Neumann M. M.,Asymptotic intertwining and spectral inclusions on Banach spaces, Czechoslovak Math. J.,43 (118) (1993), 483–497.MATHMathSciNetGoogle Scholar
  8. [8]
    Laursen K. B., Neumann M. M.,An introduction to local spectral theory, Clarendon Press, Oxford, 2000.MATHGoogle Scholar
  9. [9]
    Mbekhta M.,Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J.,29 (1987), 159–175.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Mbekhta M.,Sur la théorie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc.,110 (1990), 621–631.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Schmoeger C.,On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc.,117 (1993), 715–719.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Vasilescu F.-H.,Analytic functional calculus and spectral decompositions, Editura Academiei and D. Reidel Publishing Company, Bucharest and Dordrecht, 1982.MATHGoogle Scholar
  13. [13]
    Vrbová P.,On local spectral properties of operators in Banach spaces, Czechoslovak Math. J.,23 (98) (1973), 483–492.MathSciNetGoogle Scholar

Copyright information

© Springer 2002

Authors and Affiliations

  • T. Len Miller
    • 1
  • Vivien G. Miller
    • 1
  • Michael M. Neumann
    • 1
  1. 1.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

Personalised recommendations