대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권4호
  • /
  • Pages.893-913
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  • 2020
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  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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PROPERTIES OF OPERATOR MATRICES

  • An, Il Ju (Department of Applied Mathematics Kyung Hee University) ;
  • Ko, Eungil (Department of Mathematics Ewha Womans University) ;
  • Lee, Ji Eun (Department of Mathematics and Statistics Sejong University)
  • 투고 : 2019.06.26
  • 심사 : 2020.03.25
  • 발행 : 2020.07.01
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초록

Let 𝓢 be the collection of the operator matrices $\(\array{A&C\\Z&B}\)$ where the range of C is closed. In this paper, we study the properties of operator matrices in the class 𝓢. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class 𝓢 and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class 𝓢, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl's theorem and a-Browder's theorem, respectively.

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과제정보

The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(2017R1C1B1006538). The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(2019R1F1A1058633). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2019R1A2C1002653).

참고문헌

  1. P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht, 2004.
  2. Alatancang, G. Hou, and G. Hai, Perturbation of spectra for a class of $2{\times}2$ operator matrices, Acta Math. Appl. Sin. Engl. Ser. 28 (2012), no. 4, 711-720. https://doi.org/10.1007/s10255-012-0195-x
  3. C. Benhida, E. H. Zerouali, and H. Zguitti, Spectra of upper triangular operator matrices, Proc. Amer. Math. Soc. 133 (2005), no. 10, 3013-3020. https://doi.org/10.1090/S0002-9939-05-07812-3
  4. S. W. Brown, Hyponormal operators with thick spectra have invariant subspaces, Ann. of Math. (2) 125 (1987), no. 1, 93-103. https://doi.org/10.2307/1971289
  5. X. Cao, M. Guo, and B. Meng, Weyl's theorem for upper triangular operator matrices, Linear Algebra Appl. 402 (2005), 61-73. https://doi.org/10.1016/j.laa.2004.12.005
  6. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. http://projecteuclid.org/euclid.mmj/1031732778 https://doi.org/10.1307/mmj/1031732778
  7. R. E. Curto and Y. M. Han, Weyl's theorem for algebraically paranormal operators, Integral Equations Operator Theory 47 (2003), no. 3, 307-314. https://doi.org/10.1007/s00020-002-1164-1
  8. D. S. Cvetkovic-Ilic, An analogue to a result of Takahashi II, J. Math. Anal. Appl. 479 (2019), no. 1, 1266-1280. https://doi.org/10.1016/j.jmaa.2019.06.078
  9. S. V. Djordjevic and Y. M. Han, A note on Weyl's theorem for operator matrices, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2543-2547. https://doi.org/10.1090/S0002-9939-02-06808-9
  10. S. V. Djordjevic and H. Zguitti, Essential point spectra of operator matrices through local spectral theory, J. Math. Anal. Appl. 338 (2008), no. 1, 285-291. https://doi.org/10.1016/j.jmaa.2007.05.031
  11. J. Eschmeier and B. Prunaru, Invariant subspaces for operators with Bishop's property ($\beta$) and thick spectrum, J. Funct. Anal. 94 (1990), no. 1, 196-222. https://doi.org/10.1016/0022-1236(90)90034-I
  12. C. Ganesa Moorthy and P. S. Johnson, Composition of closed range operators, J. Anal. 12 (2004), 165-169.
  13. S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285-1315. https://doi.org/10.1090/S0002-9947-05-03742-6
  14. J. K. Han, H. Y. Lee, and W. Y. Lee, Invertible completions of $2{\times}2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000), no. 1, 119-123. https://doi.org/10.1090/S0002-9939-99-04965-5
  15. R. Harte, Invertibility and singularity for bounded linear operators, Monographs and Textbooks in Pure and Applied Mathematics, 109, Marcel Dekker, Inc., New York, 1988.
  16. I. S. Hwang and W. Y. Lee, The boundedness below of $2{\times}2$ upper triangular operator matrices, Integral Equations Operator Theory 39 (2001), no. 3, 267-276. https://doi.org/10.1007/BF01332656
  17. S. Jung, E. Ko, and J. E. Lee, On complex symmetric operator matrices, J. Math. Anal. Appl. 406 (2013), no. 2, 373-385. https://doi.org/10.1016/j.jmaa.2013.04.056
  18. S. Jung, E. Ko, and J. E. Lee, Properties of complex symmetric operators, Oper. Matrices 8 (2014), no. 4, 957-974. https://doi.org/10.7153/oam-08-53
  19. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monographs. New Series, 20, The Clarendon Press, Oxford University Press, New York, 2000.
  20. 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
  21. 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
  22. M. Oudghiri, Weyl's theorem and perturbations, Integral Equations Operator Theory 53 (2005), no. 4, 535-545. https://doi.org/10.1007/s00020-004-1342-4
  23. C. M. Pearcy, Some recent developments in operator theory, American Mathematical Society, Providence, RI, 1978.
  24. M. Putinar, Hyponormal operators are subscalar, J. Operator Theory 12 (1984), no. 2, 385-395.
  25. M. Thamban Nair, A spectral characterization of closed range operators, preprint.