Skip to main content
SpringerLink
Log in

On τ-Compactness of Products of τ-Measurable Operators

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.

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. Antonevich, A.B.: Linear functional equations. The operator approach. Universitetskoe, Minsk (1988). [in Russian]

    MATH  Google Scholar 

  2. Bikchentaev, A.M.: On noncommutative function spaces, Selected Papers in K-theory. Amer. Math. Soc. Transl. 154(2), 179–187 (1992)

    Google Scholar 

  3. Bikchentaev, A.M.: On a property of L p -spaces on semifinite von Neumann algebras. Math. Notes 64(1–2), 159–163 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bikchentaev, A.M.: Majorization for products of measurable operators, Internat. J. Theor. Phys. 37(1), 571–576 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bikchentaev, A.M.: The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra. Lobachevskii J. Math. 14, 17–24 (2004)

    MATH  MathSciNet  Google Scholar 

  6. Bikchentaev, A.M.: Commutativity of projections and trace characterization on von Neumann algebras. Sib. Math. J. 51(6), 971–977 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bikchentaev, A.M.: Commutativity of projectors and trace characterization on von Neumann algebras. II. Math. Notes 89(3–4), 461–471 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bikchentaev, A.M.: The Peierls-Bogoliubov inequality in C ∗-algebras and characterization of tracial functionals. Lobachevskii J. Math. 32(3), 175–179 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bikchentaev, A.M.: A block projection operator in normed ideal spaces of measurable operators. Russ. Math. 56(2), 75–79 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bikchentaev, A.M.: On normal τ-measurable operators affiliated with semifinite von Neumann algebras. Math. Notes 96(3–4), 332–341 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bikchentaev, A.M.: Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra. Math. Notes 98(3–4), 382–391 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bikchentaev, A.M.: On operator monotone and operator convex functions. Russ. Math. (Iz. VUZ) 60(5), 61–65 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bikchentaev, A.M.: On idempotent τ-measurable operators affiliated to a von Neumann algebra. Math. Notes 100(4), 515–525 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bikchentaev, A.: Integrable products of measurable operators. Lobachevskii J. Math. 37(4), 397–403 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bikchentaev, A.M.: Trace and integrable operators affiliated with a semifinite von Neumann algebra. Dokl. Math. 93(1), 16–19 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chu, C.: Compact product of Hankel and Toeplitz operators on the Hardy space. Indiana Univ. Math. J. 64(4), 973–982 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  17. Čučković, ž., Şahutoǧlu, S.: Compactness of products of Hankel operators on the polydisk and some product domains in ℂ 2. J. Math. Anal. Appl. 371(1), 341–346 (2010)

    Article  MathSciNet  Google Scholar 

  18. Ding, X.H.: Compact product of Toeplitz operators on the polydisk. (Chinese). Acta Math. Sinica (Chin. Ser.) 48(3), 493–498 (2005)

    MATH  MathSciNet  Google Scholar 

  19. Dodds, P.G., Dodds, T.K.-Y., de Pagter, B.: Noncommutative Köthe duality. Trans. Amer. Math. Soc. 339(2), 717–750 (1993)

  20. Fack, T., Kosaki, H.: Generalized s-numbers of τ-measurable operators. Pacific. J. Math. 123(2), 269–300 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Feshchenko, I.S.: On the closedness of the sum of ranges of operators A k with almost compact products \(A_{i}^{\ast } A_{j}\). J. Math. Anal. Appl. 416(1), 24–35 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Flores, J., Tradacete, P., Troitsky, V.G.: Disjointly homogeneous Banach lattices and compact products of operators. J. Math. Anal. Appl. 354(2), 657–663 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gohberg, I.T., Kreǐn, M.G.: Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Space. Moscow, Nauka (1965). [in Russian]

    Google Scholar 

  24. Gu, C.: Zero or compact products of several Hankel operators. Integr. Equ. Oper. Theory 43(3), 298–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. De La Harpe, P.: lnitiation à l’algèbre de Calkin. (French) Algèbres d’opèrateurs (Sèm., Les Plans-sur-Bex, 1978), pp. 180–219, Lecture Notes in Math., vol. 725. Springer, Berlin (1979)

  26. Kaftal, V.: On the theory of compact operators in von Neumann algebras. I. Indiana Univ. Math. J. 26(3), 447–457 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kaftal, V.: On the theory of compact operators in von Neumann algebras. II. Pacific. J. Math. 79(1), 129–137 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kosaki, H.: On the continuity of the map φ↦|φ| from the predual of a W ∗-algebra. J. Funct. Anal. 59(1), 123–131 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lee, Y.J., Na, K.: Compact products of Toeplitz operators on the Dirichlet space of the unit ball. J. Math. Anal. Appl. 401(2), 654–658 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15(2), 103–116 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  31. Segal, I.E.: A non-commutative extension of abstract integration. Ann. Math. 57(3), 401–457 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sukochev, F.A.: On the A. M. Bikchentaev conjecture. Russ. Math. (Iz. VUZ) 56(6), 57–59 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sukochev, F.A.: On a conjecture of A. Bikchentaev. Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy’s 60th birthday Proc. Sympos. Pure Math., 87, Amer. Math. Soc., Providence, RI, pp 327–339 (2013)

    Google Scholar 

  34. Xia, D., Zheng, D.: Compact products of Hankel operators. Integr. Equ. Oper. Theory 38(3), 357–375 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yeadon, F.J.: Non-commutative L p-spaces. Math. Proc. Cambridge Phil. Soc. 77(1), 91–102 (1975)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author is grateful to the Referee for helpful remarks.

Author information

Authors and Affiliations

  1. Kazan (Volga region) Federal University, Kazan, Russia

    Airat M. Bikchentaev

Authors
  1. Airat M. Bikchentaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Airat M. Bikchentaev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bikchentaev, A.M. On τ-Compactness of Products of τ-Measurable Operators. Int J Theor Phys 56, 3819–3830 (2017). https://doi.org/10.1007/s10773-017-3318-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-017-3318-6

Keywords

Search

Navigation