Weyl Type Theorems for Unbounded Hyponormal Operators


  • Received : 2014.10.02
  • Accepted : 2015.02.21
  • Published : 2015.09.23


If T is an unbounded hyponormal operator on an infinite dimensional complex Hilbert space H with ${\rho}(T){\neq}{\phi}$, then it is shown that T satisfies Weyl's theorem, generalized Weyl's theorem, Browder's theorem and generalized Browder's theorem. The equivalence of generalized Weyl's theorem with generalized Browder's theorem, property (gw) with property (gb) and property (w) with property (b) have also been established. It is also shown that a-Browder's theorem holds for T as well as its adjoint $T^*$.


Unbounded hyponormal operators;Weyl-type theorems;property (w);property (b)


  1. P. Aiena, Fredholm and local spectral theory with applications to multipliers, Kluwer Acad. Publishers, (2004).
  2. M. Amouch, M. Berkani, On the property (gw), Mediterr. J. Math., 5(2008), 371-378.
  3. M. Berkani, Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Amer. Math. Soc., 130(2002), 1717-1723.
  4. M. Berkani, A. Arroud, Generalized Weyl's theorem and hyponormal operators, J. Aust. Math. Soc., 76(2)(2004), 291-302.
  5. M. Berkani, N. Castro-Gonzalez, Unbounded B-Fredholm operators on Hilbert spaces, Proc. of the Edinburg Math. Soc., 51(2008), 285-296.
  6. M. Berkani, J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged), 69(2003), 359-376.
  7. M. Berkani, H. Zariouh, Extended Weyl type theorems, Mathematica Bohemica, 134(4)(2009), 369-378.
  8. L. A. Coburn, Weyl's theorem for nonnormal operators, The Michigan Mathematical Journal, 13(3)(1966), 285-288.
  9. N. Dunford, Spectral theory I. Resolution of the identity, Pacific Journal of Math., 2(1952), 559-614.
  10. J. K. Finch, The Single Valued Extension Property on a Banach Space, Pacific Journal of Math., 58 1(1975), 61-69.
  11. W. Gongbao, M. Jipu, Spectral characterization of Hyponormal Weighted Shifts, arXiv preprint math/0302275, (2003).
  12. D. C. Lay, Spectral Analysis Using Ascent, Descent, Nullity and Defect, Math. Ann., 184(1970), 197-214.
  13. V. Rakocevic, On a class of operators, Mat. Vesnik., 37(1985), 423-426.
  14. A. E. Taylor, D. C. Lay Introduction to Functional Analysis, Second Edition, New York, Wiley and sons, (1980).
  15. H. Weyl, Uber beschrankte quadatische Formen, deren Differenz Vollstetig ist, Rend. Circ. Mat. Palermo, 27(1909) 373-392.

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