Spectra of 2 × 2 Upper-Triangular Operator Matrices
Author(s)
Haiyan Zhang
Affiliation(s)
College of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China.
College of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China.
ABSTRACT
In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator 
such that 
for a given pair (A,B) of operators, where the operator 
was defined by 
. In this note, a partial answer for the question is given.
Cite this paper
H. Zhang, "Spectra of 2 × 2 Upper-Triangular Operator Matrices," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 22-25. doi: 10.4236/am.2013.411A1004.
H. Zhang, "Spectra of 2 × 2 Upper-Triangular Operator Matrices," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 22-25. doi: 10.4236/am.2013.411A1004.
References
[1] H. K. Du and J. Pan, “Perturbation of Spectrums of 2 × 2 Operator Matrices,” Proceedings of the American Mathematical Society, Vol. 121, 1994, pp. 761-766.
http://dx.doi.org/10.1090/S0002-9939-1994-1185266-2 [2] H. Y. Zhang and H. K. Du, “Browder Spectra of Upper Triangular Operator Matrices,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 700-707. http://dx.doi.org/10.1016/j.jmaa.2005.10.073 [3] M. Barrua and M. Boumazgour, “A Note on the Spectra of an Upper Triangular Operator Matrix,” Proceedings of the American Mathematical Society, Vol. 131, 2003, pp. 3083-3088. http://dx.doi.org/10.1090/S0002-9939-03-06862-X [4] X. H. Cao and B. Meng, “Essential Appoximate Point Spectra and Weyl’s Theorem for Operator Matices,” Journal of Mathematical Analysis and Applications, Vol. 304, No. 2, 2005, pp. 759-771.
http://dx.doi.org/10.1016/j.jmaa.2004.09.053 [5] D. S. Djordjevic, “Perturbations Spectra of Operator Matrices,” Journal of Operator Theory, Vol. 48, 2002, pp. 467-486. [6] J. K. Han, H. Y. Lee and W. Y. Lee, “Invertible Completions of 2 × 2 Operator Matrices,” Proceedings of the American Mathematical Society, Vol. 128, 2000, pp. 119123. http://dx.doi.org/10.1090/S0002-9939-99-04965-5 [7] Y. Li, X. H. Sun and H. K. Du, “Inversctions of the Left and Right Essential Spectra of 2 × 2 Upper Triangular Operator Matrices,” Bulletin London Mathematical Society, Vol. 36, 2004, pp. 811-819. [8] H. Y. Zhang, X. H. Zhang and H. K. Du, “Drazin Spectra of 2 × 2 Upper Triangular Operator Matrices,” Acta Mathematica Scientia, Vol. 29, 2009, pp. 272-282. [9] J. B. Conwey, “A Course in Functional Analysis,” Springer-verlag, New York, Heidelberg, Berlin, Tokyo, 1985.
http://dx.doi.org/10.1007/978-1-4757-3828-5
[1] H. K. Du and J. Pan, “Perturbation of Spectrums of 2 × 2 Operator Matrices,” Proceedings of the American Mathematical Society, Vol. 121, 1994, pp. 761-766.
http://dx.doi.org/10.1090/S0002-9939-1994-1185266-2 [2] H. Y. Zhang and H. K. Du, “Browder Spectra of Upper Triangular Operator Matrices,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 700-707. http://dx.doi.org/10.1016/j.jmaa.2005.10.073 [3] M. Barrua and M. Boumazgour, “A Note on the Spectra of an Upper Triangular Operator Matrix,” Proceedings of the American Mathematical Society, Vol. 131, 2003, pp. 3083-3088. http://dx.doi.org/10.1090/S0002-9939-03-06862-X [4] X. H. Cao and B. Meng, “Essential Appoximate Point Spectra and Weyl’s Theorem for Operator Matices,” Journal of Mathematical Analysis and Applications, Vol. 304, No. 2, 2005, pp. 759-771.
http://dx.doi.org/10.1016/j.jmaa.2004.09.053 [5] D. S. Djordjevic, “Perturbations Spectra of Operator Matrices,” Journal of Operator Theory, Vol. 48, 2002, pp. 467-486. [6] J. K. Han, H. Y. Lee and W. Y. Lee, “Invertible Completions of 2 × 2 Operator Matrices,” Proceedings of the American Mathematical Society, Vol. 128, 2000, pp. 119123. http://dx.doi.org/10.1090/S0002-9939-99-04965-5 [7] Y. Li, X. H. Sun and H. K. Du, “Inversctions of the Left and Right Essential Spectra of 2 × 2 Upper Triangular Operator Matrices,” Bulletin London Mathematical Society, Vol. 36, 2004, pp. 811-819. [8] H. Y. Zhang, X. H. Zhang and H. K. Du, “Drazin Spectra of 2 × 2 Upper Triangular Operator Matrices,” Acta Mathematica Scientia, Vol. 29, 2009, pp. 272-282. [9] J. B. Conwey, “A Course in Functional Analysis,” Springer-verlag, New York, Heidelberg, Berlin, Tokyo, 1985.
http://dx.doi.org/10.1007/978-1-4757-3828-5