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Positivity

, Volume 19, Issue 4, pp 911–925 | Cite as

On Fuglede–Putnam properties

  • Seonjin Jo
  • Yoenha Kim
  • Eungil Ko
Article

Abstract

In this paper, we study the Fuglede–Putnam property. We give a necessary and sufficient condition for which \((\oplus _{i=1}^{n}A_i,\oplus _{i=1}^{n}B_i)\) satisfies the Fuglede–Putnam property. We also study the local spectral theory associated with the Fuglede–Putnam property. Finally, we define the weak Fuglede–Putnam property and we investigate several cases which satisfy the weak Fuglede–Putnam property.

Keywords

Fuglede–Putnam theorem Commutator Generalized commutator Aluthge transform Normal operator 

Mathematics Subject Classification

47A50 47A63 

Notes

Acknowledgments

The authors wish to thank the referee for a careful reading and valuable comments for the original draft.

References

  1. 1.
    Aluthge, A.: On \(p\)-hyponormal operators for 0 \(<\) p \(<\) 1. Integr. Equ. Oper. Theory 13, 307–315 (1990)Google Scholar
  2. 2.
    Aluthge, A., Wang, D.: An operator inequality which implies paranormality. Math. Inequal. Appl. 2, 113–119 (1999)MATHMathSciNetGoogle Scholar
  3. 3.
    Aluthge, A., Wang, D.: w-Hyponormal operators. Integr. Equ. Oper. Theory 36, 1–10 (2000)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bachir, A.: Fuglede–Putnam theorem for w-hyponormal or class \({{\cal Y}}\)-operators. Ann. Funct. Anal. 4, 53–60 (2013)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Berberian, S.: Extensions of a theroem of Fuglede–Putnam. Proc. Am. Math. Soc. 71, 113–114 (1978)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Colojoara, I., Foias, C.: Theory of Generalized Spectral Operators. Gordon and Breach, New York (1968)MATHGoogle Scholar
  7. 7.
    Conway, J.B.: A Course in Functional Anaylsis. Springer Verlag, New York (1990)Google Scholar
  8. 8.
    Fuglede, B.: A commutativity theorem for normal operators. Proc. Natl. Acad. Sci. 36, 35–40 (1950)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gao, F., Fang, X.: The Fuglede–Putnam theorem and Putnam’s inequaility for quasi-class \((A, k)\) operators. Ann. Funct. Anal. 2, 105–113 (2011)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Garcia, S.R., Wogen, W.R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362, 6065–6077 (2010)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Jeon, I., Tanahashi, K., Uchiyama, A.: On quasisimilartiy for \(\log \)-hyponormal operators. Glasg. Math. J. 46, 169–176 (2004)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Jung, I., Ko, E., Pearcy, C.: Aluthge transform of operators. Integr. Equ. Oper. Theory 37, 437–448 (2000)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Laursen, K., Neumann, M.: An Introduction to Local Spectral Theory. Clarendon Press, Oxford (2000)MATHGoogle Scholar
  14. 14.
    Mecheri, S.: Why we solve the operator equation \(AX - XB = C\), preprintGoogle Scholar
  15. 15.
    Mecheri, S., Tanahashi, K., Uchiyama, A.: Fuglede–Putnam theorem for p-hyponormal or class \({{\cal Y}}\) operators. Bull. Korean Math. Soc. 43, 747–753 (2006)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Moslehian, M.S., Nabavi Sales, S.M.S.: Fuglede–Putnam type theorems via the Aluthge transform. Positivity 17, 151–162 (2013)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Putnam, C.R.: On normal operators in Hilbert space. Am. J. Math. 73, 357–362 (1951)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Radjabalipour, M.: An extension of Putnam–Fuglede theorem for hyponormal operators. Math. Z. 194, 117–120 (1987)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Radjavi, H., Rosenthal, P.: Invariant Subspaces. Dover Publications Inc., Mineola (2003)MATHGoogle Scholar
  20. 20.
    Takahashi, K.: On the converse of the Fuglede–Putnam theorem. Acta Sci. Math. (Szeged) 43, 123–125 (1981)MATHMathSciNetGoogle Scholar
  21. 21.
    Tanahashi, K.: On \(\log \)-hyponormal operators. Integr. Equ. Oper. Theory 34, 364–372 (1999)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsEwha Womans UniversitySeoulKorea
  2. 2.Institute of Mathematical SciencesEwha Womans UniversitySeoulKorea

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