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Properties of Linear Relations in Banach Spaces

  • Guojing RenEmail author
  • Ravi P. Agarwal
Article
  • 22 Downloads

Abstract

This paper is concerned with properties and stability of linear relations, including closed, compact, and Fredholm linear relations. Stability of closedness and compactness of linear relations under relatively bounded perturbations and relatively compact perturbations is studied. In particular, results on the stability of index of Fredholm linear relation and its iterates are discussed. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.

Keywords

Linear relation Perturbation Closedness Compactness Fredholm 

Mathematics Subject Classification

47A06 47A53 47A55 

Notes

References

  1. 1.
    Abdmouleh, F., Alvarez, T., Ammar, A., Jeribi, A.: Spectral mapping theorem for Rakocevic and Schmoeger essential spectra of a multivalued linear operator. Mediterr. J. Math. 12, 1019–1031 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alvarez, T., Ammar, A., Jeribi, A.: On the essential spectra of some matrix of linear relations. Math. Methods Appl. Sci. 37(5), 620–644 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Azizov, T., Dijksma, A., Wanjala, G.: Compressions of maximal dissipative and self-adjoint linear relations and of dilations. Linear Algebra Appl. 439, 771–792 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baskakov, A.G., Zagorskii, A.S.: Spectral theory of linear relations on real Banach spaces. Mat. Zametki 81, 17–31 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chafai, E., Mnif, M.: Perturbation of normally solvable linear relations in paracomplete spaces. Linear Algebra Appl. 439, 1875–1885 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chaker, W., Jeribi, A., Krichen, B.: Demicompact linear operators, essential spectrum and some perturbation results. Math. Nachr. 288(13), 1476–1486 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coddington, E.A.: Extension theory of formally normal and symmetric linear relations. Mem. Am. Math. Soc. 134 (1973)Google Scholar
  9. 9.
    Cross, R.: Multivalued Linear Operators. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  10. 10.
    Goldberg, S.: Unbounded Linear Operators: Theory and Applications. Dover Publications Inc, New York (2006)zbMATHGoogle Scholar
  11. 11.
    Jeribi, A., Moalla, N.: A characterization of some subsets of Schechter’s essential spectrum and application to singular transport equation. J. Math. Anal. Appl. 358, 434–444 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kato, T.: Perturbation Theory for nullity, deficiency and other quantities of Linear Operators. J. D’Analyse Math. 6, 261–322 (1958)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)zbMATHGoogle Scholar
  14. 14.
    Naimark, M.A.: Linear Differential Operators, Part II, Linear Differential Operators in Hilbert Space. Ungar Publ. Co., New York (1968)zbMATHGoogle Scholar
  15. 15.
    Nagy, B.S.: On the stability of the index of unbounded linear transformations. Acta. Math. Acad. Sci. Hunger. 3, 49–51 (1952)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ren, G.: On the density of the minimal subspaces generated by discrete linear Hamiltonian systems. Appl. Math. Lett. 27, 1–5 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ren, G.: Stability of index for linear relations and its applications. Indagationes Mathematicae (2017), In PressGoogle Scholar
  18. 18.
    Sandovici, A., de Snoo, H., Winkler, H.: Ascent, descent, nullity, defect and related notions for linear relations in linear spaces. Linear Algebra Appl. 423, 456–497 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shi, Y., Shao, C., Ren, G.: Spectral properties of self-adjoint subspaces. Linear Algebra Appl. 438, 191–218 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shi, Y., Xu, G., Ren, G.: Boundedness and closedness of subspaces of product spaces. Linear Multilinear A. 66(2), 309–333 (2018)CrossRefGoogle Scholar
  21. 21.
    Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, New York (1980)CrossRefGoogle Scholar
  22. 22.
    Yood, B.: Properties of linear transformations preserved under addition of a completely continous transformation. Duke Math. J. 18, 599–612 (1951)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wilcox, D.L.: Essential spectra of linear relations. Linear Algebra Appl. 462, 110–125 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and Quantitative EconomicsShandong University of Finance and EconomicsJinanPeople’s Republic of China
  2. 2.Department of MathematicsTexas AM University-KingsvilleKingsvilleUSA

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