Skip to main content
Log in

Essentially self-adjoint linear relations in Hilbert spaces

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R. Arens, Operational calculus of linear relations. Pacific J. Math. 11, 9–23 (1961)

    Article  MathSciNet  Google Scholar 

  2. E.A. Coddington, H.S.V. de Snoo, Positive self-adjoint extensions of positive symmetric subspaces. Math. Z. 159, 203–214 (1978)

    Article  MathSciNet  Google Scholar 

  3. A. Favini, A. Yagi, Degenerate Differential Equations in Banach Spaces (Marcel Dekker, New York, 1999)

    MATH  Google Scholar 

  4. S. Hassi, On the Friedrichs and the Kreĭn-von Neumann extension of nonnegative relations. Acta Wasaensia 122, 37–54 (2004)

    MATH  Google Scholar 

  5. S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, Form sums of nonnegative self-adjoint operators. Acta Math. Hungar. 111, 81–105 (2006)

    Article  MathSciNet  Google Scholar 

  6. S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, A general factorization approach to the extension theory of nonnegative operators and relations. J. Operator Theory 58, 351–386 (2007)

    MATH  Google Scholar 

  7. S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, Extremal extensions for the sum of nonnegative self-adjoint relations. Proc. Am. Math. Soc. 135, 3193–3204 (2007)

    Article  Google Scholar 

  8. S. Hassi, H.S.V. de Snoo, Factorization, majorization, and domination for linear relations. Ann. Univ. Sci. Budapest 58, 55–72 (2015)

    MathSciNet  MATH  Google Scholar 

  9. S. Hassi, H.S.V. de Snoo, F.H. Szafraniec, Componentwise and canonical decompositions of linear relations. Dissertationes Mathematicae 465, 59 (2009)

    Article  Google Scholar 

  10. T. Kato, Perturbation Theory for Linear Operators. Corrected Printing of the Second Edition (Springer, Berlin, 1980)

    Google Scholar 

  11. M. Roman, A. Sandovici, The square root of nonnegative self-adjoint linear relations in Hilbert spaces. J. Oper. Theory 82(2), 357–367 (2019)

    MATH  Google Scholar 

  12. A. Sandovici, On polynomials in a linear relation, Proceedings of the 4th Workshop on Operator Theory in Krein Spaces and Applications, Berlin, December 17-19, 2004. In volume Operator Theory in Inner Product Spaces, Series: OTAA, Vol. 175, Förster, K.-H.; Jonas, P.; Langer, H.; Trunk, C. (Eds.) 2007, VI, 240 p., ISBN: 978-3-7643-8269-8, 231–240

  13. A. Sandovici, On domains and ranges of powers of linear relations in linear spaces. Complex Anal. Oper. Theory 6(3), 749–758 (2012)

    Article  MathSciNet  Google Scholar 

  14. A. Sandovici, Von Neumann’s theorem for linear relations. Linear Multilinear Algebra 66(9), 1750–1756 (2018)

    Article  MathSciNet  Google Scholar 

  15. A. Sandovici, Self-adjointness and skew-adjointness criteria involving powers of linear relations. J. Math. Anal. Appl. 470, 186–200 (2019)

    Article  MathSciNet  Google Scholar 

  16. K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265 (Springer, Dordrecht, 2012)

    Book  Google Scholar 

  17. Z. Sebestyén, J. Stochel, Restrictions of positive self-adjoint operators. Acta Sci. Math. (Szeged) 55, 149–154 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Z. Sebestyén, Zs Tarcsay, \(T^{*}T\) always has a positive self-adjoint extension. Acta Math. Hungar. 135, 116–129 (2012)

    Article  MathSciNet  Google Scholar 

  19. Z. Sebestyén, Zs Tarcsay, Characterizations of self-adjoint operators. Stud. Sci. Math. Hungar. 50, 423–435 (2013)

    MATH  Google Scholar 

  20. Z. Sebestyén, Zs Tarcsay, Characterizations of essentially self-adjoint and skew-adjoint operators. Stud. Sci. Math. Hungar. 52(3), 371–385 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Z. Sebestyén, Zs Tarcsay, Adjoint of sums and products of operators in Hilbert spaces. Acta Sci. Math. (Szeged) 82(1–2), 175–191 (2016)

    Article  MathSciNet  Google Scholar 

  22. Z. Sebestyén, Zs Tarcsay, Operators having self-adjoint squares. Annales Univ. Sci. Budapest. Sect. Math. 58, 105–110 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewer for his/her thoughtful comments and efforts towards improving the present paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Adrian Sandovici.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roman, M., Sandovici, A. Essentially self-adjoint linear relations in Hilbert spaces. Period Math Hung 83, 122–132 (2021). https://doi.org/10.1007/s10998-020-00373-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-020-00373-8

Keywords

Mathematics Subject Classification

Navigation