## Abstract and Applied Analysis

### The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

#### Abstract

When $A\in B\left(H\right)$ and $B\in B\left(K\right)$ are given, we denote by ${M}_{C}$ the operator acting on the infinite-dimensional separable Hilbert space $H\oplus K$ of the form ${M}_{C}=\left(\begin{smallmatrix}A& C\\[5pt] \mathrm{0}& B\end{smallmatrix}\right)$. In this paper, it is proved that there exists some operator $C\in B\left(K,H\right)$ such that ${M}_{C}$ is upper semi-Browder if and only if there exists some left invertible operator $C\in B\left(K,H\right)$ such that ${M}_{C}$ is upper semi-Browder. Moreover, a necessary and sufficient condition for ${M}_{C}$ to be upper semi-Browder for some $C\in G\left(K,H\right)$ is given, where $G\left(K,H\right)$ denotes the subset of all of the invertible operators of $B\left(K,H\right)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 373147, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512029

Digital Object Identifier
doi:10.1155/2013/373147

Mathematical Reviews number (MathSciNet)
MR3096835

Zentralblatt MATH identifier
1318.47007

#### Citation

Zhang, Shifang; Zhong, Huaijie; Long, Long. The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices. Abstr. Appl. Anal. 2013 (2013), Article ID 373147, 8 pages. doi:10.1155/2013/373147. https://projecteuclid.org/euclid.aaa/1393512029

#### References

• X. H. Cao, “Browder spectra for upper triangular operator matrices,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 477–484, 2008.
• X. H. Cao and B. Meng, “Essential approximate point spectra and Weyl's theorem for operator matrices,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 759–771, 2005.
• X. L. Chen, S. F. Zhang, and H. J. Zhong, “On the filling in holes problem for operator matrices,” Linear Algebra and its Applications, vol. 430, no. 1, pp. 558–563, 2009.
• D. S. Djordjević, “Perturbations of spectra of operator matrices,” Journal of Operator Theory, vol. 48, no. 3, pp. 467–486, 2002.
• H. K. Du and P. Jin, “Perturbation of spectrums of $2\times 2$ operator matrices,” Proceedings of the American Mathematical Society, vol. 121, no. 3, pp. 761–766, 1994.
• S. V. Djordjević and Y. M. Han, “Spectral continuity for operator matrices,” Glasgow Mathematical Journal, vol. 43, no. 3, pp. 487–490, 2001.
• G. J. Hai and A. Chen, “Perturbations of the right and left spectra for operator matrices,” Journal of Operator Theory, vol. 67, no. 1, pp. 207–214, 2012.
• J. K. Han, H. Y. Lee Youl, and W. Y. Lee Young, “Invertible completions of 2 $\times$ 2 upper triangular operator matrices,” Proceedings of the American Mathematical Society, vol. 128, no. 1, pp. 119–123, 2000.
• I. S. Hwang and W. Y. Lee, “The boundedness below of $2\times 2$ upper triangular operator matrices,” Integral Equations and Operator Theory, vol. 39, no. 3, pp. 267–276, 2001.
• W. Y. Lee, “Weyl's theorem for operator matrices,” Integral Equations and Operator Theory, vol. 32, no. 3, pp. 319–331, 1998.
• W. Y. Lee, “Weyl spectra of operator matrices,” Proceedings of the American Mathematical Society, vol. 129, no. 1, pp. 131–138, 2001.
• M. Z. Kolundžija and D. S. Djordjević, “Generalized invertibility of operator matrices,” Arkiv för Matematik, vol. 50, no. 2, pp. 259–267, 2012.
• Y. Li and H. Du, “The intersection of essential approximate point spectra of operator matrices,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1171–1183, 2006.
• E. H. Zerouali and H. Zguitti, “Perturbation of spectra of operator matrices and local spectral theory,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 992–1005, 2006.
• S. F. Zhang, Z. Y. Wu, and H. J. Zhong, “Continuous spectrum, point spectrum and residual spectrum of operator matrices,” Linear Algebra and its Applications, vol. 433, no. 3, pp. 653–661, 2010.
• S. F. Zhang and H. J. Zhong, “A note on Browder spectrum of operator matrices,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 927–931, 2008.
• S. F. Zhang, H. J. Zhong, and Q. F. Jiang, “Drazin spectrum of operator matrices on the Banach space,” Linear Algebra and its Applications, vol. 429, no. 8-9, pp. 2067–2075, 2008.
• S. F. Zhang, H. J. Zhong, and J. D. Wu, “Spectra of $2\times 2$ upper-triangular operator matrices,” Acta Mathematica Sinica, vol. 54, no. 1, pp. 41–60, 2011 (Chinese).
• S. F. Zhang, H. J. Zhong, and J. D. Wu, “Fredholm perturbation of spectra of $2\times 2$-upper triangular matrices,” Acta Mathematica Sinica, vol. 54, no. 4, pp. 581–590, 2011 (Chinese).
• S. F. Zhang and J. D. Wu, “Samuel multiplicities and Browder spectrum of operator matrices,” Operators and Matrices, vol. 6, no. 1, pp. 169–179, 2012.
• Y. N. Zhang, H. J. Zhong, and L. Q. Lin, “Browder spectra and essential spectra of operator matrices,” Acta Mathematica Sinica (English Series), vol. 24, no. 6, pp. 947–954, 2008.
• S. F. Zhang and Z. Q. Wu, “Characterizations of perturbations of spectra of $2\times 2$ upper triangular operator matrices,” Journal of Mathematical Analysis and Applications, vol. 392, no. 2, pp. 103–110, 2012.
• H. Zguitti, “On the Drazin inverse for upper triangular operator matrices,” Bulletin of Mathematical Analysis and Applications, vol. 2, no. 2, pp. 27–33, 2010.
• X. Fang, “Samuel multiplicity and the structure of semi-Fredholm operators,” Advances in Mathematics, vol. 186, no. 2, pp. 411–437, 2004.
• P. A. Fillmore and J. P. Williams, “On operator ranges,” Advances in Mathematics, vol. 7, pp. 254–281, 1971.