대한수학회지 (Journal of the Korean Mathematical Society)

  • 제54권1호
  • /
  • Pages.281-302
  • /
  • 2017
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED BROWDER, WEYL SPECTRA AND THE POLAROID PROPERTY UNDER COMPACT PERTURBATIONS

  • Duggal, Bhaggy P. (8 Redwood Grove) ;
  • Kim, In Hyoun (Department of Mathematics Incheon National University)
  • 투고 : 2015.12.02
  • 발행 : 2017.01.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

For a Banach space operator $A{\in}B(\mathcal{X})$, let ${\sigma}(A)$, ${\sigma}_a(A)$, ${\sigma}_w(A)$ and ${\sigma}_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator A is polaroid (resp., left polaroid), if the points $iso{\sigma}(A)$ (resp., $iso{\sigma}_a(A)$) are poles (resp., left poles) of the resolvent of A. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A{\in}B(\mathcal{X})$, we prove that a sufficient condition for: (i) A+K to have SVEP on the complement of ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) for every compact operator $K{\in}B(\mathcal{X})$ is that ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) has no holes; (ii) A + K to be polaroid (resp., left polaroid) for every compact operator $K{\in}B(\mathcal{X})$ is that iso${\sigma}_w(A)$ = ∅ (resp., $iso{\sigma}_{aw}(A)$ = ∅). It is seen that these conditions are also necessary in the case in which the Banach space $\mathcal{X}$ is a Hilbert space.

키워드

참고문헌

  1. P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer, 2004.
  2. P. Aiena and V. Muller, The localized single-valued extension property and Riesz operators, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2051-2055. https://doi.org/10.1090/S0002-9939-2014-12404-X
  3. P. Aiena and J. E. Sanabria, On left and right poles of the resolvent, Acta Sci. Math. (Szeged) 74 (2008), no. 3-4, 669-687.
  4. A. Amouch and H. Zguitti, On the equivalence of Browder's and generalized Browder's theorem, Glasgow Math. J. 48 (2006), no. 1, 179-185. https://doi.org/10.1017/S0017089505002971
  5. C. Apostol, L. A. Fialkow, D .A. Herrero, and D. Voiculescu, Approximation of Hilbert Space Operators, Vol. II, Research Notes in Mathematics 102, Pitman Advanced Publishing Program, 1984.
  6. M. Berkani and J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Math. Sci. (Szeged) 69 (2003), no. 1-2, 359-376.
  7. B. P. Duggal, Hereditarily normaloid operators, Extracta Math. 20 (2005), no. 2, 203-217.
  8. B. P. Duggal, Polaroid operators, SVEP and perturbed Browder, Weyl theorems, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 3, 317-330. https://doi.org/10.1007/BF03032085
  9. B. P. Duggal, SVEP, Browder and Weyl Theorems, Topicas de Theoria de la Approximacion III. In:Jimenez Pozo, M. A., Bustamante Gonzalez, J., Djordjevic, S. V. (eds.) Textos Cientificos BUAP Puebla, pp. 107-146; Freely available at http://www.fcfm.buap.mx/CA/analysis-mat/pdf/LIBRO-TOP-T-APPROX.pdf
  10. B. P. Duggal, I. H. Jeon, and I. H. Kim, Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems, J. Inequal. Appl. 2013 (2013), 268, 12pp. https://doi.org/10.1186/1029-242X-2013-268
  11. S. Grabiner, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), no. 2, 317-337. https://doi.org/10.2969/jmsj/03420317
  12. D. A. Herrero, Approximation of Hilbert Space Operators, Vol. I, Research Notes in Mathematics 72, Pitman Advanced Publishing Program, 1982.
  13. D. A. Herrero, Economical compact perturbations, J. Operator Theory 19 (1988), no. 1, 25-42.
  14. D. A. Herrero, T. J. Taylor, and Z. Y. Wang, Variation of the point spectrum under compact perturbations, Topics in operator theory, 113-158, Oper. Theory Adv. Appl., 32, Birkhauser, Basel, 1988.
  15. H. G. Heuser, Functional Analysis, John Wiley & Sons, 1982.
  16. Y. Q. Ji, Quasitriangular + small compact = strongly irreducible, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4657-4673. https://doi.org/10.1090/S0002-9947-99-02307-7
  17. C. S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, Birkhauser, Boston, 2012.
  18. K. B. Laursen and M. M. Neumann, Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000.
  19. C. G. Li and T. T. Zhou, Polaroid type operators and compact perturbations, Studia Math. 221 (2014), no. 2, 175-192. https://doi.org/10.4064/sm221-2-5
  20. V. Rakocevic, Semi-Browder operators and perturbations, Studia Math. 122 (1997), no. 2, 131-137. https://doi.org/10.4064/sm-122-2-131-137
  21. A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, 1980.
  22. S. Zhu and C. G. Li, SVEP and compact perturbations, J. Math. Anal. Appl. 380 (2011), no. 1, 69-75. https://doi.org/10.1016/j.jmaa.2011.02.036