Advertisement

On the Joint Numerical Spectrum in Banach Spaces

  • Wissal Boubaker
  • Nedra Moalla
  • Agnes Radl
Original Paper
  • 3 Downloads

Abstract

The purpose of this paper is to introduce the joint numerical spectrum of a q tuple of operators on a Banach space and to study its properties. This notion generalizes both the joint numerical range and the numerical spectrum.

Keywords

Numerical spectrum Joint spectrum Joint numerical range 

Mathematics Subject Classification

Primary 47A10 Secondary 47A12 

Notes

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript.

References

  1. 1.
    Adler, M., Dada, W., Radl, A.: A semigroup approach to the numerical range of operators on Banach spaces. Semigroup Forum 94, 51–70 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonsall, F.F., Duncan, J.: Numerical ranges of operators on normed spaces and elements of normed algebras. Cambridge (1971)Google Scholar
  3. 3.
    Bonsall, F.F., Duncan, J.: Numerical Range II. Cambridge University Press, Cambridge (1973)CrossRefzbMATHGoogle Scholar
  4. 4.
    Buoni, J.J., Wadhwa, B.L.: On joint numerical ranges. Pacific J. Math 77, 303–306 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cho, M., Takaguchi, M.: Boundary points of joint numerical ranges. Pacific J. Math. 95, 27–35 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dada, W.: A Semigroup Approach to Numerical Ranges of Operators on Banach Spaces. PhD Thesis, University of Tübingen (2014)Google Scholar
  7. 7.
    Dash, A.T.: Joint numerical range. Glasnik Mat. 7, 75–81 (1972)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dash, A.T.: Joint spectra. Stud. Math. 45, 225–237 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Engel, K.J., Nagel, R.: One Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  10. 10.
    Gustafson, K.E., Rao, D.K.M.: Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext, Springer, New York (1997)Google Scholar
  11. 11.
    Gutkin, E., Jonckheere, E.A., Karow, M.: Convexity of the joint numerical range: topological and differential geometric viewpoints. Linear Algebra Appl. 376, 143–171 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huang, D.: Joint numerical ranges for unbounded normal operators. Proc. Edinburgh Math. Soc. 28, 225–232 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, C.K., Poon, Y.T.: Convexity of the joint numerical range. SIAM J. Matrix Anal. Appl. 21, 668–678 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Müller, V.: The joint essential numerical range, compact perturbations and the Olsen problem. Stud. Math. 197, 275–290 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wrobel, V.: Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces. Glasgow Math. J. 30, 145–153 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang, Y.O.: Joint spatial numerical ranges of operator on Banach spaces. Bull. Korean Math. Soc. 26, 119–126 (1989)MathSciNetzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences of SfaxUniversity of SfaxSfaxTunisia
  2. 2.Mathematisches InstitutUniversität LeipzigLeipzigGermany

Personalised recommendations