Advertisement

The Closed Extensions of a Closed Operator

  • Christoph FischbacherEmail author
Article
  • 12 Downloads

Abstract

Given a densely defined and closed (but not necessarily symmetric) operator A acting on a complex Hilbert space \({\mathcal {H}}\), we establish a one-to-one correspondence between its closed extensions and subspaces \({\mathfrak {M}}\subset {\mathcal {D}}(A^*)\), that are closed with respect to the graph norm of \(A^*\) and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of \(A^*\). After this, we will express our results using the language of Gel’fand triples generalizing the well-known results for the selfadjoint case.

Keywords

Closed extensions Gelfand triples Symmetric and selfadjoint operators 

Notes

Acknowledgements

I am very grateful to Sergii Kuzhel for valuable feedback on this manuscript. I would also like to thank him, Yury Arlinskiĭ and the referee for providing me with useful references. Parts of this work have been done during my PhD studies at the University of Kent in Canterbury, UK (cf. [11, Chapter 4]), and I would also like to thank my thesis advisors Sergey Naboko and Ian Wood for support and guidance. Finally, I express my appreciation to the UK Engineering and Physical Sciences Research Council (Doctoral Training Grant Ref. EP/K50306X/1) and the School of Mathematics, Statistics and Actuarial Science at the University of Kent for a Ph.D. studentship.

References

  1. 1.
    Albeverio, S., Kuzhel, S., Nizhnik, L.: Singularly perturbed self-adjoint operators in scales of Hilbert spaces (in Russian). Ukraïn. Mat. Zh. 59(6), 723–743 (2007). (English transl. in Ukrainian Math. J. 59 (2007), 787–810)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Kuzhel, S., Nizhnik, L.: On the perturbation theory of self-adjoint operators. Tokyo J. Math. 31(2), 273–292 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alonso, A., Simon, B.: The Birman–Kreĭn–Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4, 251–270 (1980)zbMATHGoogle Scholar
  4. 4.
    Arlinskiĭ, Yu.: On m-accretive extensions and restrictions. Methods Funct. Anal. Topol. 4, 1–26 (1998)MathSciNetGoogle Scholar
  5. 5.
    Arlinskiĭ, Yu.: M-accretive extensions of sectorial operators and Krein spaces. Oper. Theory Adv. Appl. 118, 67–82 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Arlinskiĭ, Yu.: Boundary triplets and maximal accretive extensions of sectorial operators. In: Hassi, S., de Snoo, H.S.V., Szafraniec, F.H. (eds.) Operator Methods for Boundary Value Problems, 1st edn, pp. 35–72. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  7. 7.
    Arlinskiĭ, Yu., Tsekanovskiĭ, E.: Some remarks on singular perturbations of self-adjoint operators. Methods Funct. Anal. Topol. 9, 287–308 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Arlinskiĭ, Y., Tsekanovskiĭ, E.: M. Kreĭn’s research on semi-bounded operators, its contemporary developments and applications. Oper. Theory Adv. Appl. 190, 65–112 (2009)zbMATHGoogle Scholar
  9. 9.
    Berezanskii, YuM: Expansions in Eigenfunctions of Selfadjoint Operators. American Mathematical Society, Providence (1968)CrossRefGoogle Scholar
  10. 10.
    Crandall, M., Phillips, R.: On the extension problem for dissipative operators. J. Funct. Anal. 2, 147–176 (1968)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fischbacher, C.: On the Theory of Dissipative Extensions, Ph.D. Thesis, University of Kent, (2017). https://kar.kent.ac.uk/61093/. Accessed 11 July 2019
  12. 12.
    Grubb, G.: A characterization of the non-local boundary value problems associated with an elliptic operator, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze \(3^e\) série 22, 425–513 (1968)Google Scholar
  13. 13.
    Hassi, S., Kuzhel, S.: On symmetries in the theory of finite rank singular perturbations. J. Funct. Anal. 256, 777–809 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)CrossRefGoogle Scholar
  15. 15.
    Kiselev, A., Simon, B.: Rank-one perturbations at infinitesimal coupling. J. Funct. Anal. 128, 245–252 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kurasov, P., Kuroda, S.T.: Krein’s resolvent formula and perturbation theory. J. Oper. Theory 51(2), 321–334 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kuroda, S.T., Nagatani, H.: Resolvent formulas of general type and its application to point interactions. J. Evol. Equ. 1, 421–440 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kuzhel, S., Znojil, M.: Non-self-adjoint Schrödinger operators with nonlocal one-point interactions. Banach J. Math. Anal. 11(4), 923–944 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lieb, E., Loss, M.: Analysis, Second Edition, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)Google Scholar
  20. 20.
    Phillips, R.S.: On dissipative operators. Lect. Differ. Equ. 3, 65–113 (1969)Google Scholar
  21. 21.
    Posilicano, A.: A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sebestyén, Z., Stochel, J.: On suboperators with codimension one domains. J. Math. Anal. Appl. 360, 391–397 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil I Grundlagen. Verlag B.G, Teubner, Stuttgart (2000). (German)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations