Skip to main content
SpringerLink
Log in

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arlinskiĭ, Yu., Tsekanovskiĭ, E.: On von Neumann’s problem in extension theory of nonnegative operators. Proc. Am. Math. Soc. 131, 3143–3154 (2003)

    Article  MATH  Google Scholar 

  2. Bender, C.M., Boettcher, S.: Real spectra in non-hermitian hamiltonians having \(\mathcal{P}\mathcal{T}\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  4. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  5. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Foulis, D.J.: Effects, observables, states, and symmetries in physics. Found. Phys. 37, 1421–1446 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Hedlíková, J., Pulmannová, S.: Generalized difference posets and orthoalgebras. Acta Math. Univ. Comen. LXV, 247–279 (1996)

    Google Scholar 

  8. Kalmbach, G., Riečanová, Z.: An axiomatization for abelian relative inverses. Demonstr. Math. 27, 769–780 (1994)

    MATH  Google Scholar 

  9. Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Paseka, J.: \(\mathcal{P}\mathcal{T}\)-symmetry in (generalized) effect algebras. Int. J. Theor. Phys. 50, 1198–1205 (2011). doi:10.1007/s10773-010-0594-9

    Article  MATH  MathSciNet  Google Scholar 

  11. Polakovič, M., Riečanová, Z.: Generalized effect algebras of positive operators densely defined on Hilbert space. Int. J. Theor. Phys. 50, 1167–1174 (2011). doi:10.1007/s10773-010-0458-3

    Article  MATH  Google Scholar 

  12. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

    MATH  Google Scholar 

  13. Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras. Int. J. Theor. Phys. 38, 3209–3220 (1999)

    Article  MATH  Google Scholar 

  14. Riečanová, Z.: Effect algebras of positive self-adjoint operators densely defined on Hilbert spaces. Acta Polytech. (Proceedings of the 7-th DI Microconference Analytic and Algebraic Methods VII, Prague, March 2011, ed. V. Jakubský and M. Znojil). to appear

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Paseka

  2. Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19, Bratislava, Slovak Republic

    Zdenka Riečanová

Authors
  1. Jan Paseka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zdenka Riečanová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Paseka.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paseka, J., Riečanová, Z. Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras. Found Phys 41, 1634–1647 (2011). https://doi.org/10.1007/s10701-011-9573-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-011-9573-0

Keywords

Search

Navigation