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Browder spectra for upper triangular operator matrices. (English) Zbl 1139.47006

Summary: When $A\in B\left(H\right)$ and $B\in B\left(K\right)$ are given, we denote by ${M}_{C}$ an operator of the form ${M}_{C}=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$ acting on the infinite-dimensional separable Hilbert space $H\oplus K$. In this paper, we prove that

$\bigcap _{C\in B\left(K,H\right)}{\sigma }_{b}\left({M}_{C}\right)={\sigma }_{ab}\left(A\right)\cup {\sigma }_{ab}\left({B}^{*}\right)\cup \left\{\lambda \in ℂ:n\left(A-\lambda I\right)+n\left(B-\lambda I\right)\ne d\left(A-\lambda I\right)+d\left(B-\lambda I\right)\right\},$

where ${\sigma }_{b}\left(T\right)$, ${\sigma }_{ab}\left(T\right)$, $n\left(T\right)$ and ${T}^{*}$ denote the Browder spectrum, Browder essential approximate point spectrum, nullity, deficiency and conjugate of $T$, respectively. Some related results are obtained.

##### MSC:
 47A10 Spectrum and resolvent of linear operators 47A53 (Semi-)Fredholm operators; index theories
##### References:
 [1] Berberian, S. K.: An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan math. J. 16, 273-279 (1969) · Zbl 0175.13603 · doi:10.1307/mmj/1029000272 [2] Berberian, S. K.: The Weyl spectrum of an operator, Indiana univ. Math. J. 20, 529-544 (1970) · Zbl 0203.13401 · doi:10.1512/iumj.1970.20.20044 [3] Cao, X. H.; Guo, M. Z.; Meng, B.: Semi-Fredholm spectrum and Weyl’s theorem for operator matrices, Acta math. Sinica 22, 169-178 (2006) · Zbl 1129.47014 · doi:10.1007/s10114-004-0505-1 [4] Cao, X. H.; Meng, B.: Essential approximate point spectrum and Weyl’s theorem for operator matrices, J. math. Anal. appl. 304, 759-771 (2005) · Zbl 1083.47006 · doi:10.1016/j.jmaa.2004.09.053 [5] Coburn, L. A.: Weyl’s theorem for nonnormal operator, Michigan math. J. 13, 285-288 (1966) · Zbl 0173.42904 · doi:10.1307/mmj/1031732778 [6] Djordjević, D. S.: Perturbations of spectra of operator matrices, J. operator theory 48, 467-486 (2002) · Zbl 1019.47003 [7] Hong-Ke, Du; Jin, Pan: Perturbation of spectrums of $2×2$ operator matrices, Proc. amer. Math. soc. 121, 761-766 (1994) · Zbl 0814.47016 · doi:10.2307/2160273 [8] Han, J. K.; Lee, H. Y.; Lee, W. Y.: Invertible completions of $2×2$ upper triangular operator matrices, Proc. amer. Math. soc. 128, 119-123 (2000) · Zbl 0944.47004 · doi:10.1090/S0002-9939-99-04965-5 [9] Lee, W. Y.: Weyl spectra of operator matrices, Proc. amer. Math. soc. 129, 131-138 (2000) [10] Lee, W. Y.: Weyl’s theorem for operator matrices, Integral equations operator theory 32, 319-331 (1998) · Zbl 0923.47001 · doi:10.1007/BF01203773 [11] Oberai, K. K.: On the Weyl spectrum, Illinois J. Math. 18, 208-212 (1974) · Zbl 0277.47002 [12] Weyl, H.: Über beschränkte quadratische formen, deren differenz vollstetig ist, Rend. circ. Mat. Palermo 27, 373-392 (1909) · Zbl 40.0395.01