Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEYL@S THEOREMS FOR POSINORMAL OPERATORS

  • Published : 2005.05.01
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Abstract

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.

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References

  1. P. Aiena and O. Monsalve, The single valued extension property and the generalized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), 461-477
  2. P. Aiena and F. Villafane, Weyl's theorem of some classes of operators, Extracta Math.(in press)
  3. P. Aiena and M. Mbekhta, Characterization of some classes of operators by means of the Kato decomposition, Boll. Unione. Mat. Ital. 10-A(1966), 609-621
  4. S. R. Caradus, W. E. Pfaffenberger and Y. Bertram, Calkin Algebras and Algebras of operators on Banach Spaces, Marcel Dekker, New York, 1974
  5. I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968
  6. R. E. Curto and Y. M. Han, Weyl's theorem, a-Weyl's theorem and local spectral theory, J. London Math. Soc. 67 (2003), 499-509 https://doi.org/10.1112/S0024610702004027
  7. B. P. Duggal, A remark on generalized Putnam-Fuglede theorems, Proc. Amer. Math. Soc. 129 (2000), 83-87 https://doi.org/10.1090/S0002-9939-00-05920-7
  8. B. P. Duggal, Weyl's theorem for a generalized derivation and an elementary operator, Mat. Vesnik 54 (2002), 71-81
  9. B. P. Duggal, S. V. Djordjvic, and C. S. Kubrusly, Kato type operators and Weyl's theorem, J. Math. Anal. Appl.(in press)
  10. M. R. Embry and M. Rosenblum, Spectra, tensor products and linear operator equations, Pacific J. Math. 53 (1974), 95-107 https://doi.org/10.2140/pjm.1974.53.95
  11. J. Eschmeier and M. Putinar, Bishop's condition ($\beta$) and rich extensions of linear operators, Indiana Univ. Math. J. 37 (1988), 325-347 https://doi.org/10.1512/iumj.1988.37.37016
  12. R. E. Harte and W. Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), 2115-2124 https://doi.org/10.1090/S0002-9947-97-01881-3
  13. H. G. Heuser, Functional Analysis, John Wiley and Sons, 1982
  14. In Ho Jeon, Se Hee Kim, Eungil Ko, and Ji Eum Park, On positive-normal operators, Bull. Korean Math. Soc. 39 (2002), 33-41 https://doi.org/10.4134/BKMS.2002.39.1.033
  15. T. Kato, Perturbation theory for nullity, defiency and other quantities of linear operators, J. Math. Anal. 6 (1958), 261-322 https://doi.org/10.1007/BF02790238
  16. W. H. Lee and W. Y. Lee, A spectral mapping theorem for the Weyl spectrum, Glasg. Math. J. 38 (1996), 61-64 https://doi.org/10.1017/S0017089500031268
  17. K. B. Laursen and M. N. Neumann, Introduction to local spectral theory, Clarendon Press, Oxford, 2000
  18. M. Mbekhta, Generalisation de la decomposition de Kato aux operateurs paranormaux et spectraux, Glasg. Math. J. 29 (1987), 159-175 https://doi.org/10.1017/S0017089500006807
  19. M. Oudghiri, Weyl's and Browder's theorem for operators satisfying the SVEP, Studia Math. 163 (2004), 85-101 https://doi.org/10.4064/sm163-1-5
  20. M. Radjabalipour, An extension of Putnam-Fuglede theorem for hyponormal operators, Math. Z. 194 (1987), 117-120 https://doi.org/10.1007/BF01168010
  21. V. Rakocevic, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl. 34 (1989), 915-919
  22. H.'C. Rhaly Jr., Posinormal operators, J. Math. Soc. Japan 46 (1994), 587-605 https://doi.org/10.2969/jmsj/04640587

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