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Perturbation of spectra for a class of 2×2 operator matrices

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Abstract

In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M X = ( A X0 B ) and M Z = ( A C Z B ) on the Hilbert space H ⊕ K and the sets \(\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} \) and \(\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} \), where R(C) is a closed subspace, are characterized

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References

  1. Alatancang, Huang, J., Fan, X. The residual spectrum for a class of infinite dimensional Hamiltonian operators in L 2 × L 2. Acta Mathematiea Scientia, 25(7): 1040–1045 (2005) (in Chinese)

    MathSciNet  Google Scholar 

  2. Alatancang, Huang, J., Fan, X. Structure of the spectrum of infinite dimensional Hamiltonian operators. Science in China Series A-Mathematics, 51(5): 915–924 (2008)

    Article  MathSciNet  Google Scholar 

  3. Alatancang; Wu, D. Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator. Science in China Series A-Mathematics, 52(1): 173–180 (2009)

    Article  MathSciNet  Google Scholar 

  4. Barraa, M., Boumazgour, M. A note on the spectrum of an upper triangular operator matrices. Proceedings of the American Mathematical Society, 131(10): 3083–3088 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conway, J.B. A course in functional analysis, second edition. Springer Verlag, New York, 1990

    MATH  Google Scholar 

  6. Cui, J., Hou, J. On the spectral completion of operator partial matrices. Acta Mathematica Sinica, 42(1): 181–186 (1999) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  7. Djordjevi’c, D.S. Perturbations of spectra of operator matrices. Journal of Operator Theory, 48(3): 467–486 (2002)

    MathSciNet  Google Scholar 

  8. Du, H., Pan, J. Perturbation of spectrums of 2 × 2 operator matrices. Proceedings of the American Mathematical Society, 121(3): 761–766 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hai G., Chen, A. The semi-Fredholmness of 2 ×2 operator matrices. Journal of Mathematical Analysis and Applications, 352(2): 733–738 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hai G., Chen, A. On the right (left) invertible completions for operator matrices. Integral Equations Operator Theory 67(1): 79–93 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Han, J., Lee, H., Lee, W. Invertible completions of 2 × 2 upper triangular operator matrices. Proceedings of the American Mathematical Society, 128: 119–123 (1999)

    Article  MathSciNet  Google Scholar 

  12. Hou, G., Alatancang. Perturbation of spectrums of 2×2 upper triangular operator matrices. Journal of Systems Science and Mathematical Sciences, 26(3): 257–263 (2006) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  13. Hou, G., Alatancang. Completeness of eigenfunction systems for off-diagonal infinite-dimensional Hamiltonian operators. Communications in Theoretical Physics, 53(2): 237–241 (2010)

    Article  MATH  Google Scholar 

  14. Hou, G., Alatancang. Spectra of off-diagonal infinite-dimensional Hamiltonian operators and their applications to plane elasticity problems. Communications in Theoretical Physics, 51(2): 200–204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hou, G., Alatancang, Huang, J. Invertibility of an operator appearing in the linear-quadratic optimal control problems in a Hilbert space. Acta Mathematica Sinica, 50(2): 473–480 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Huang, J., Alatancang, Wu, H. Descriptions of spectra of infinite dimensional Hamiltonian operators and their applications. Mathematische Nachrichten, 283(8): 1144–1154 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Y., Sun, X., Du, H. The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices. Linear Algebra and its Applications, 418(1): 112–121 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Müller, V. Spectral theory of linear operators. Birkhäuser Verlag, Berlin, 2003

    MATH  Google Scholar 

  19. Ren, F., Du, H., Cao, H. The intersection of the spectra of operator completions. Linear Algebra and its Applications, 371: 103–109 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sun, J., Wang, Z. Spectral analysis of linear operators. Science Press, Beijing, 2005

    Google Scholar 

  21. Wu, D., Chen, A. Spectral inclusion properties of the numerical range in a space with an indefinite metric. Linear Algebra and its Applications, 435(5): 1131–1136 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, D., Chen, A. Invertibility of nonnegative Hamiltonian operator with unbounded entries. Journal of Mathematical Analysis and Applications, 373(2): 410–413 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Alatancang, Guo-lin Hou & Guo-jun Hai

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  1. Alatancang
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  2. Guo-lin Hou
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  3. Guo-jun Hai
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Corresponding author

Correspondence to Guo-jun Hai.

Additional information

Supported by the National Natural Science Foundation of China (No. 10962004), Tianyuan Fund for Mathematics (No. 11126307), the National Natural Science Foundation of Inner Mongolia (No. 2011MS0104, 2012MS0105), the Research Program of Science at Universities of Inner Mongolia Autonomous Region (No. NJZZ11011) and Program of Higher-level Talents of Inner Mongolia University (No. Z20100116).

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Alatancang, Hou, Gl. & Hai, Gj. Perturbation of spectra for a class of 2×2 operator matrices. Acta Math. Appl. Sin. Engl. Ser. 28, 711–720 (2012). https://doi.org/10.1007/s10255-012-0195-x

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