Title: A Spectral Analysis of Linear Operator Pencils on Banach Spaces with Application to Quotient of Bounded Operators
Author(s): Bekkai Messirdi, Abdellah Gherbi, Mohamed Amouch
Pages: 104-128
Cite as:
Bekkai Messirdi, Abdellah Gherbi, Mohamed Amouch, A Spectral Analysis of Linear Operator Pencils on Banach Spaces with Application to Quotient of Bounded Operators, Int. J. Anal. Appl., 7 (2) (2015), 104-128.

Abstract


Let X and Y two complex Banach spaces and (A,B) a pair of bounded linear operators acting on X with value on Y. This paper is concerned with spectral analysis ofthe pair (A;B): We establish some properties concerning the   spectrum of the linear operator pencils (A-lambda B) when B is not necessarily invertible and lambda is a complex number. Also, we use the functional calculus for the pair (A,B) to prove the corresponding spectral mapping theorem for (A,B). In addition, we define the generalized Kato essential spectrum and the closed range spectra of the pair (A,B) and we give some relationships between this spectrums. As application, we describe a spectral analysis of quotient operators.

Full Text: PDF

 

References


  1. F. Abdmouleh, A. Ammar and A. Jeribi, Stability of the S-Essential Spectra on a Banach Space, Math. Slovaca, 63, 2 (2013), 1-22.

  2. F. Aguirre and C. Conca, Eigenfrequencies of a Tube Bundle Immersed in a Fluid, Appl. Math. Optim., 18, (1988), 1-38.

  3. P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Pub., 2004.

  4. R. I. Andrushkiw, On the Spectral Theory of Operator Pencils in a Hilbert Space, Nonlinear Math. Phys., 2, 3/4 (1995), 356-366.

  5. W.G. Bade, An operational calculus for operators with spectrum in a strip, Pacific J. Math. 3, (1953), 257-290.

  6. M. Benharrat and B. Messirdi, On the generalized Kato spectrum, Serdica Math. J., 37, (2011), 283-294.

  7. M. Benharrat and B. Messirdi, Relationship between the Kato essential spectrum and a variant of essential spectrum, Gen. Math. Rev., 20, 4 (2012), 71-88.

  8. M. Benharrat and B. Messirdi, Essential spectrum a brief survey of concepts and applications, Azerb. J. Math., 2, 1 (2012), 35-61.

  9. M. Sh. Birman, A. Laptev, Discrete spectrum of the perturbed Dirac operator, Mathematical results in quantum mechanics (Blossin, 1993), 55-59. Oper. Theory Adv. Appl., 70, Birkhauser, Basel, 1994.

  10. M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, Vestn. Leningr. Univ., 1, (1977), 13-21.

  11. F. E. Browder, On the spectral theory of elliptic differential operators, I, Math. Ann. 142,(1961), 22-130.

  12. J. Bronski, M. Johnson and T. Kapitula, An instability index theory for quadratic pencils and applications, Comm. Math. Physics 327, 2 (2014), 521-550.

  13. D. Chu and G. H. Golub, On a Generalized Eigenvalue Problem for Nonsquare Pencils, SIAM J. Matrix Anal. Appl., 28, 3 (2006), 770-787.

  14. V. V. Ditkin, Certain spectral properties of a pencil of linear bounded operators, Mathematical notes of the Academy of Sciences of the USSR, 31, 1 (1982), 39-41.

  15. M. Faierman, R. Mennicken and M. Moller, A boundary eigenvalue problem for a system of partial differential operators occuring in magnetohydrodynamics, Math. Nachr., 173, (1995), 141–167.

  16. Gestztesy G., Gurarie D., H. Holden, M. Klaus, L. Sadun, B. Simon and P. Vogl, Trapping and cascading of eigenvalues in the large coupling limit, Commun. Math. Phys. 118, (1988), 597-634.

  17. A. Gherbi, B. Messirdi and M. Benharrat, On the quotient of two bounded operators, Submitted, june 2014.

  18. R.R.Hartmann, N.J.Robinson and M.E.Portnoi Smooth electron waveguides in graphene, Phys. Rev. B, 81, 24 (2010), 245-431.

  19. E. Hille and R. C. Phillips, Functional Analysis and Semi-Groups, Russian translation, IL, Moscow (1962).

  20. A. Jeribi, N. Moalla and S. Yengui, S-essential spectra and application to an example of transport operators, Math. Methods Appl. Sci., 37 (2012), 2341-2353.

  21. Q. Jiang and H. Zhong, Generalized Kato decomposition, single-valued extension property and approximate point spectrum, J. Math. Anal. Appl. 356, (2009) 322-327.

  22. T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, (1995).

  23. N. Khaldi, M. Benharrat and B. Messirdi, On the spectral boundary value problems and boundary approximate controllability of linear systems, Rend. Circ. Mat. Palermo, 63, (2014),141– 153.

  24. D. Lay, Characterizations of the essential spectrum of F. E. Browder, Bull. Amer. Math. Soc., 74, (1968), 246–248.

  25. M. Mbekhta and A. Ouahab, Op´erateur s-r´egulier dans un espace de Banach et th´eorie spectrale, Pub. Irma. Lille. Vol. 22, (1990), No XII.

  26. V. Muller, On the regular spectrum, J. Operator Theory, 31, (1994), 363-380.

  27. V. Rakoˇcevi´c, Generalized spectrum and commuting compact perturbations, Proc. Ed-inb. Math. Soc. 36, (1993), 197-209.

  28. F. G. Ren, X. M. Yang, Some properties on the spectrum of linear operator pencils, Basic Sci. J. of Textile Universities, 25, 2 (2012), 127-131.

  29. H. H. Rosenbrok, State Space and Multivariable Theory, Thomas Nelson, London (1970).

  30. M. Sahin, M. B. Ragimov, Spectral Theory of Ordered Pairs of the Linear Operators - acting in Different Banach Spaces and Applications, Internat. Math. Forum, 2, 5 (2007), 223 - 236.

  31. D.A.Stone, C.A.Downing, and M.E.Portnoi Searching for confined modes in graphene channels: The variable phase method, Phys. Rev. B, 86, 7 (2012), 075464.

  32. C. Tretter, Linear oerator pencils (A − λB) with discrete spectrum, Integral Equations Oper. Theory, 317, (2000) 127-141.

  33. C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, (2008).

  34. K. Yosida, Functional Analysis, Springer-Verlag Berlin, (1965).