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Journal of Mathematical Sciences

, Volume 240, Issue 1, pp 1–20 | Cite as

Solvable Extensions of Some Nondensely Defined Operators and the Resolvents of These Extensions

  • О. H. Storozh
Article
  • 15 Downloads

In terms of abstract boundary conditions, we study a class of extensions of finite-dimensional restrictions of closed densely defined linear operators acting in Hilbert spaces. By the methods of the theory of linear relations, we find the resolvent sets and construct the resolvents of the analyzed extensions. The set of these extensions is parameterized by a certain auxiliary operator. In the case where this operator is normally solvable, we present certain improvements of the basic results.

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References

  1. 1.
    M. I. Vishik, “On the general boundary-value problems for elliptic differential equations,” Trudy Mosk. Mat. Obshch., 1, 187–246 (1952).Google Scholar
  2. 2.
    V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Operator Differential Equations, Springer, Berlin (1991).CrossRefzbMATHGoogle Scholar
  3. 3.
    T. Kato, Theory of the Perturbations of Linear Operators [Russian translation), Mir, Moscow (1972).Google Scholar
  4. 4.
    A. N. Kochubei, “The extensions of a nondensely defined symmetric operator,” Sib. Mat. Zh., 18, No. 2, 314–320 (1977); English translation: Sib. Math. J., 18, No. 2, 225–229 (1977).Google Scholar
  5. 5.
    M. A. Krasnosel’skii, “On self-adjoint extensions of Hermitian operators,” Ukr. Mat. Zh., No. 1, 21–38 (1949).Google Scholar
  6. 6.
    V. E. Lyantse and O. G. Storozh, Methods of the Theory of Unbounded Operators [in Russian], Naukova Dumka, Kiev (1983).Google Scholar
  7. 7.
    М. M. Malamud, “On one approach to the theory of extensions of nondensely defined Hermitian operators,” Dokl. Acad. Nauk Ukr. SSR, No. 3, 20–25 (1990).Google Scholar
  8. 8.
    O. H. Storozh, “Relationship between two couples of linear relations and dissipative extensions of some nondensely defined operators,” Karpat. Mat. Publik., 1, No. 2, 207–213 (2009).zbMATHGoogle Scholar
  9. 9.
    O. H. Storozh, Methods of the Theory of Extensions and Differential-Boundary Operators [in Ukrainian], Doctoral-Degree Thesis (Physics and Mathematics), Lviv (1995).Google Scholar
  10. 10.
    A. V. Štraus, “On the extensions and the characteristic function of a symmetric operator,” Izv. Acad. Nauk SSSR, Ser. Mat., 32, No. 1, 186–207 (1968); English translation: Math. USSR-Izv., 2, No. 1, 181–203 (1968).Google Scholar
  11. 11.
    R. Arens, “Operational calculus of linear relations,” Pacific J. Math., 11, No. 1, 9–23 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyén, and H. S. V. de Snoo, “On the class of extremal extensions of a nonnegative operator,” in: L. Kérchy, I. Gohberg, C. I. Foias, and H. Langer (editors), Recent Advances in Operator Theory and Related Topics, Operator Theory: Advances and Applications, Vol. 127, 41–81 (2001).Google Scholar
  13. 13.
    V. M. Bruk, “On the characteristic operator of an integral equation with a Nevanlinna measure in the infinite-dimensional case,” Zh. Mat. Fiz. Anal. Geom., 10, No. 2, 163–188 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E. A. Coddington, “Self-adjoint subspace extensions of nondensely defined symmetric operators,” Bull. Amer. Math. Soc., 79, No. 4, 712–715 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    H. S. V. de Snoo, V. A. Derkach, S. Hassi, and М. M. Malamud, “Generalized resolvents of symmetric operators and admissibility,” Methods Funct. Anal. Topol., 6, No. 3, 24–55 (2000).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. Dijksma and H. S. V. de Snoo, “Self-adjoint extensions of symmetric subspaces,” Pacific J. Math., 54, No. 1, 71–100 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    S. Hassi, H. S. V. de Snoo, A. E. Sterk, and H. Winkler, “Finite-dimensional graph perturbations of self-adjoint Sturm–Liouville operators,” in: Operator Theory, Structured Matrices, and Dilations. Tiberiu Constantinescu Memorial Volume, Theta Foundation, Bucharest (2007), pp. 205–226.Google Scholar
  18. 18.
    S. Hassi, H. S. V. de Snoo, and F. H. Szafraniec, “Infinite-dimensional perturbations, maximally nondensely defined symmetric operators, and some matrix representations,” Indag. Math., 23, No. 4, 1087–1117 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. V. Kuzhel and S. A. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht (1998).zbMATHGoogle Scholar
  20. 20.
    M. M. Malamud and V. I. Mogilevskii, “On extensions of dual pairs of operators,” Dopov. Nats. Acad. Nauk Ukr., No. 1, 30–37 (1997).Google Scholar
  21. 21.
    Iu. I. Oliiar and O. G. Storozh, “On a criterion of mutual adjointness for the extensions of some nondensely defined operators,” Methods Funct. Anal. Topol., 20, No. 1, 50–58 (2014).MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • О. H. Storozh
    • 1
  1. 1.I. Franko Lviv National UniversityLvivUkraine

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