Fuglede–Putnam theorem for \((\alpha , \beta )\)-normal operators

Rendiconti del Circolo Matematico di Palermo Series 2 volume 69, pages1243–1249(2020)Cite this article

Abstract

In this paper, we obtain some properties of \((\alpha , \beta )\)-normal and we prove following assertions.

  1. (i)

    If T is \((\alpha , \beta )\)-normal operator, S is an invertible operator and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\).

  2. (ii)

    If T is totally \((\alpha , \beta )\)-normal operator, then the range of generalized derivation \(\delta _{T}: \mathcal {B}( {\mathcal H}) \ni X \rightarrow TX- XT \in \mathcal {B}( {\mathcal H}) \) is orthogonal to its kernel.

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

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. 1.

    Berberian, S.K.: Extensions of a theorem of Fuglede and Putnam. Proc. Am. Math. Soc. 71, 113–114 (1978)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990)

    Google Scholar 

  3. 3.

    Bachir, A., Lombarkia, F.: Fuglede–Putnam’s theorem for w-hyponormal operators. Math. Inequal. Appl. 15, 777–786 (2012)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bachir, A.: Fuglede–Putnam theorem for \(w\)-hyponormal or class \({\cal{Y}}\) operators. Ann. Funct. Anal. 4(1), 53–60 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Berberian, S.K.: Approximate proper vectors. Proc. Am. Math. Soc. 13, 111–114 (1962)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Brown, A., Percy, C.: Spectra of tensor products of operators. Proc. Am. Math. Soc. 17, 162–166 (1966)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dragomir, S.S., Moslehian, M.S.: Some inequalities for \((\alpha , \beta )\)-normal operators in Hilbert spaces. Series Mathematics and Informatics, pp. 39–47 (2008)

  8. 8.

    Douglas, R.G.: On majorization, factorization and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17, 413–415 (1966)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Furuta, T.: An extenstion of the Fuglede theorem to subnormal operators using a Hilbert–Schmidt norm inequality. Proc. Am. Math. Soc. 81, 240–242 (1981)

    Article  Google Scholar 

  10. 10.

    Hou, J.C.: On tensor products of operators. Acta Math. Sin. 91, 195–202 (1993)

    MATH  Google Scholar 

  11. 11.

    Moore, R.L., Rogers, D.D., Trent, T.T.: A note on intertwining M-hyponormal operators. Proc. Am. Math. Soc. 83, 514–516 (1981)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Mecheri, S.: Finite operators and orthogonality. Nihonkai Math. 19, 53–60 (2008)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mecheri, S.: A generalization of Fuglede–Putnam theorem. J. Pure Math. 21, 31–38 (2004)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Moslehian, M.S.: On \((\alpha , \beta )\)-normal operators in Hilbert spaces, pp. 39–47. Problem, IMAGE (2007)

  15. 15.

    Senthilkumar, D., Sherin Joy, S.M.: On totally \((\alpha, \beta )\)-normal operators. Far East J. Math. Sci. (FJMS) 71(1), 151–167 (2012)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Pagacz, P.: The Putnam–Fuglede property for paranormal and *-paranormal operators. Opusc. Math. 33(3), 565–574 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Putnam, C.R.: On the normal operators on Hilbert space. Am. J. Math. 73, 357–362 (1951)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Rashid, M.H.M., Noorani, M.S.M.: On relaxation normality in the Fulgede–Putnam’s theorem for a quasi-class \({\cal{A}}\) operators. Tamkang. J. Math. 40, 307–312 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Stochel, J.: Seminormality of operators from their tensor product. Proc. Am. Math. Soc. 124, 435–440 (1996)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Uchiyama, A., Tanahashi, K.: Fuglede–Putnam’s theorem for p-hyponormal or log-hyponormal operators. Glasgow Math. J. 44, 397–410 (2002)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Takahashi, K., Uchiyama, M.: Every \(C_{00}\) contractions with Hilbert-Schmidt defect operator is of class \(C_{0}\). J. Oper. Theory 12, 331–335 (1983)

    MATH  Google Scholar 

  22. 22.

    Yoshino, T.: Remark on the generalized Putnam–Fuglede theorem. Proc. Am. Math. Soc. 95, 571–572 (1985)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Williams, J.P.: Finite operators. Proc. Am. Math. Soc. 26, 129–135 (1970)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Wielandt, H.: Helmut ber die Unbeschrnktheit der Operatoren der Quantenmechanik, (German). Math. Ann. 121, 21 (1949)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

  1. Department of Mathematics, King Khalid University, Abha, Saudi Arabia

    A. Bachir

  2. Department of Mathematics, Cochin University of Science and Technology, Cochin-22, India

    T. Prasad

Authors
  1. A. Bachir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bachir.

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

Verify currency and authenticity via CrossMark

Cite this article

Bachir, A., Prasad, T. Fuglede–Putnam theorem for \((\alpha , \beta )\)-normal operators. Rend. Circ. Mat. Palermo, II. Ser 69, 1243–1249 (2020). https://doi.org/10.1007/s12215-019-00454-9

Download citation

Keywords

  • Fuglede–Putnam theorem
  • Hyponormal operator
  • \((\alpha</Keyword> <Keyword>\beta )\)-Normal operator
  • Orthogonality

Mathematics Subject Classification

  • 47B20
  • 47A10