Skip to main content
SpringerLink
Log in

On Means of Nonnegative Sesquilinear Forms

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The aim of this paper is to generalize the theory of operator connections for nonnegative sesquilinear forms. As an application, we investigate the case of bounded finitely additive set functions. One of the most important connections in this setting is the parallel sum. We introduce this notion, and in addition, we present a Lebesgue-type decomposition theorem for such functions.

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. Anderson W. N. Jr., Morley T. D., Trapp G. E.: Characterization of parallel subtraction. Proc. Nat. Acad. Sci. U.S.A. 76, 3599–3601 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson W. N. Jr., Duffin R. J.: Series and parallel addition of matrices. J. Math. Anal. Appl. 26, 576–594 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ando T.: Lebesgue-type decomposition positive operators. Acta. Sci. Math. (Szeged) 38, 253–260 (1976)

    MATH  MathSciNet  Google Scholar 

  4. T. Ando and F. Kubo, Means of positive linear operators, Math. Ann., 246 (1979/80), 205–224.

    Google Scholar 

  5. T. Ando, Topics on Operator Inequalities, Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University (Sapporo, 1978).

  6. Asplund E.: Averaged norms. Israel J. Math. 5, 227–233 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fujii J. I.: On geometric and harmonic means of positive operators. Math. Japon 24, 203–207 (1979)

    MATH  MathSciNet  Google Scholar 

  8. Hassi S., Sebestyén Z., de Snoo H.: Lebesgue type decompositions for nonnegative forms. J. Funct. Anal. 257, 3858–3894 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Hassi, Z. Sebestyén and H. de Snoo, Domain and range descriptions for adjoint relations, and parallel sums and differences of forms, in: Recent Advances in Operator Theory in Hilbert and Krein Spaces, Oper. Theory Adv. Appl. (198), Birkhäuser Verlag (Basel, 2010), pp. 211–227.

  10. Morley T. D.: Parallel summation, Maxwell’s principle and the infimum of projections. J. Math. Anal. Appl. 70, 33–41 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nishio N.: Characterization of Lebesgue-type decomposition of positive operators. Acta Sci. Math. (Szeged) 42, 143–152 (1980)

    MATH  MathSciNet  Google Scholar 

  12. Pekarev È.L., Šmul’jan Ju. L.: Parallel addition and parallel subtraction of operators. Izv. Akad. Nauk SSSR Ser. Mat. 40, 366–387 (1976)

    MathSciNet  Google Scholar 

  13. Pusz W., Woronowicz S. L.: Functional calculus for sesquilinear forms and the purification map. Rep. Mathematical Phys. 8, 159–170 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sebestyén Z., Tarcsay Zs., Titkos T.: Lebesgue decomposition theorems. Acta Sci. Math. (Szeged) 79, 219–233 (2013)

    MATH  MathSciNet  Google Scholar 

  15. Sebestyén Z., Titkos T.: Complement of forms. Positivity 17, 1–15 (2013)

    MATH  MathSciNet  Google Scholar 

  16. Sebestyén Z., Titkos T.: Erratum to: Complement of forms. Positivity 17, 941–943 (2013)

    MATH  MathSciNet  Google Scholar 

  17. Sebestyén Z., Titkos T.: Parallel subtraction of nonnegative forms. Acta Math. Hungar. 136, 252–269 (2012)

    MATH  MathSciNet  Google Scholar 

  18. Szűcs Zs.: On the Lebesgue decomposition of positive linear functionals. PAMS 41, 619–623 (2013)

    Google Scholar 

  19. Tarcsay Zs.: Lebesgue-type decomposition of positive operators. Positivity 17, 803–817 (2013)

    MATH  MathSciNet  Google Scholar 

  20. Zs. Tarcsay, A functional analytic proof of the Lebesgue–Darst decomposition theorem, Real Analysis Exchange, 39 (2013/2014), 241–248.

  21. Zs. Tarcsay, Lebesgue decomposition of representable functionals on *-algebras, manuscript, submitted.

  22. Titkos T.: Ando’s theorem for nonnegative forms. Positivity 16, 619–626 (2012)

    MATH  MathSciNet  Google Scholar 

  23. Titkos T.: Lebesgue decomposition of contents via nonnegative forms. Acta Math. Hungar. 140, 151–161 (2013)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117, Budapest, Hungary

    T. Titkos

Authors
  1. T. Titkos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Titkos.

Additional information

Dedicated to Zoltán Sebestyén on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Titkos, T. On Means of Nonnegative Sesquilinear Forms. Acta Math. Hungar. 143, 515–533 (2014). https://doi.org/10.1007/s10474-014-0416-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-014-0416-2

Key words and phrases

Mathematics Subject Classification

Search

Navigation