Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WEYL'S THEOREM AND TENSOR PRODUCT FOR OPERATORS SATISFYING T*k|T2|Tk≥T*k|T|2Tk

  • Kim, In-Hyoun (Department of Mathematics, University of Incheon)
  • Published : 2010.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

For a bounded linear operator T on a separable complex infinite dimensional Hilbert space $\mathcal{H}$, we say that T is a quasi-class (A, k) operator if $T^{*k}|T^2|T^k\;{\geq}\;T^{*k}|T|^2T^k$. In this paper we prove that if T is a quasi-class (A, k) operator and f is an analytic function on an open neighborhood of the spectrum of T, then f(T) satisfies Weyl's theorem. Also, we consider the tensor product for quasi-class (A, k) operators.

Keywords

References

  1. T. Ando, Operators with a norm condition, Acta Sci. Math. (Szeged) 33 (1972), 169–178.
  2. S. K. Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273–279. https://doi.org/10.1307/mmj/1029000272
  3. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288. https://doi.org/10.1307/mmj/1031732778
  4. B. P. Duggal, Weyl's theorem for totally hereditarily normaloid operators, Rend. Circ. Mat. Palermo (2) 53 (2004), no. 3, 417–428. https://doi.org/10.1007/BF02875734
  5. B. P. Duggal, I. H. Jeon, and I. H. Kim, On Weyl's theorem for quasi-class A operators, J. Korean Math. Soc. 43 (2006), no. 4, 899–909. https://doi.org/10.4134/JKMS.2006.43.4.899
  6. T. Furuta, M. Ito, and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. Math. 1 (1998), no. 3, 389–403
  7. R. E. Harte, Invertibility and Singularity for Bounded Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics, 109. Marcel Dekker, Inc., New York, 1988.
  8. J.-C. Hou, On the tensor products of operators, Acta Math. Sinica (N.S.) 9 (1993), no. 2, 195–202. https://doi.org/10.1007/BF02560050
  9. I. H. Jeon and B. P. Duggal, On operators with an absolute value condition, J. Korean Math. Soc. 41 (2004), no. 4, 617–627. https://doi.org/10.4134/JKMS.2004.41.4.617
  10. I. H. Jeon and I. H. Kim, On operators satisfying $T^{\ast}|T^2|T{\geq}T^{\ast}|T|^2T$, Linear Algebra Appl. 418 (2006), no. 2-3, 854–862. https://doi.org/10.1016/j.laa.2006.02.040
  11. I. H. Kim, Tensor products of log-hyponormal operators, Bull. Korean Math. Soc. 42 (2005), no. 2, 269–277.
  12. W. Y. Lee, Weyl's theorem for operator matrices, Integral Equations Operator Theory 32 (1998), no. 3, 319–331. https://doi.org/10.1007/BF01203773
  13. W. Y. Lee, Weyl spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001), no. 1, 131–138 https://doi.org/10.1090/S0002-9939-00-05846-9
  14. W. Y. Lee and S. H. Lee, A spectral mapping theorem for the Weyl spectrum, Glasgow Math. J. 38 (1996), no. 1, 61–64. https://doi.org/10.1017/S0017089500031268
  15. T. Saito, Hyponormal operators and related topics, Lectures on operator algebras (dedicated to the memory of David M. Topping; Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. II), pp. 533–664. Lecture Notes in Math., Vol. 247, Springer, Berlin, 1972.
  16. J. Stochel, Seminormality of operators from their tensor product, Proc. Amer. Math. Soc. 124 (1996), no. 1, 135–140. https://doi.org/10.1090/S0002-9939-96-03017-1
  17. K. Tanahashi, I. H. Jeon, I. H. Kim, and A. Uchiyama, Quasinilpotent part of class A or (p, k)-quasihyponormal operators (preprint).
  18. A. Uchiyama, Weyl's theorem for class A operators, Math. Inequal. Appl. 4 (2001), no. 1, 143–150.
  19. A. Uchiyama and S. V. Djordjevic, Weyls theorem for p-quasihyponormal operators (preprint).
  20. H. Weyl, Uber beschrankte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392. https://doi.org/10.1007/BF03019655

Cited by

  1. WEYL'S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS An*OPERATO vol.51, pp.5, 2014, https://doi.org/10.4134/JKMS.2014.51.5.1089