Li, Yuan; Du, Hongke The intersection of essential approximate point spectra of operator matrices. (English) Zbl 1108.47013 J. Math. Anal. Appl. 323, No. 2, 1171-1183 (2006). Suppose that \({\mathcal K}\) and \({\mathcal H}\) are separable Hilbert spaces. Let \(B({\mathcal K},{\mathcal H})\), \(B_l({\mathcal K},{\mathcal H})\), and \(\text{Inv}({\mathcal K},{\mathcal H})\) be the sets of all bounded linear operators, left invertible bounded linear operators, and invertible bounded linear operators from \({\mathcal K}\) to \({\mathcal H}\), respectively. Let \(\Phi_+\) be the set of all upper semi-Fredholm operators. Consider the sets \(\Phi_+^-=\{T\in\Phi_+:\text{ind}(T)\leq 0\}\) and \(\sigma_{ea}(T)=\{\lambda\in{\mathbb C}:T-\lambda\notin\Phi_+^-\}\). Suppose that \(A\in B({\mathcal H})\) and \(B\in B({\mathcal K})\) are fixed. For \(C\in B({\mathcal K},{\mathcal H})\), \(M_C=\left(\begin{smallmatrix} A & C \\ 0 & B \end{smallmatrix}\right)\in B({\mathcal H}\oplus{\mathcal K})\). In the present paper, the sets \(\bigcap_{C\in B_l({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\), \(\bigcap_{C\in \text{Inv}({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\), and \(\bigcup_{C\in \text{Inv}({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\) are characterized. In particular, \[ \bigcap_{C\in\text{Inv}({\mathcal K},{\mathcal H})} \sigma_{ea}(M_C) = \bigcap_{C\in B({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C) \cup \{\lambda\in{\mathbb C}: B-\lambda\;\text{is compact}\}. \] Reviewer: Alexei Yu. Karlovich (Braga) Cited in 14 Documents MSC: 47A53 (Semi-) Fredholm operators; index theories 47A10 Spectrum, resolvent 47A55 Perturbation theory of linear operators Keywords:semi-Fredholm operator; left essential spectrum; right essential spectrum; index; defect numbers; \(2\times 2\) operator matrix; perturbations of spectra PDFBibTeX XMLCite \textit{Y. Li} and \textit{H. Du}, J. Math. Anal. Appl. 323, No. 2, 1171--1183 (2006; Zbl 1108.47013) Full Text: DOI References: [1] Conway, J. B., A Course in Functional Analysis (1989), Springer: Springer New York [2] Cao, X. H.; Meng, B., Essential approximate point spectra and Weyl’s theorem for operator matrices, J. Math. Anal. Appl., 304, 759-771 (2005) · Zbl 1083.47006 [3] Djordjevic, D. S., Perturbations of spectra of operator matrices, J. Operator Theory, 48, 467-486 (2002) · Zbl 1019.47003 [4] Djordjevic, S. V.; Han, Y. M., Browder’s theorems and spectral continuity, Glasg. Math. J., 42, 479-486 (2000) · Zbl 0979.47004 [5] Du, H. K.; Pan, J., Perturbations of spectra of \(2 \times 2\) operator matrices, Proc. Amer. Math. Soc., 121, 761-767 (1994), MR 94i:47004 [6] Fillmore, P. A.; Willams, J. P., On operator ranges, Adv. Math., 7, 254-281 (1971) · Zbl 0224.47009 [7] Hwang, I. S.; Lee, W. Y., The boundedness below of \(2 \times 2\) upper triangular operator matrices, Integral Equations Operator Theory, 39, 267-276 (2001) · Zbl 0986.47004 [8] Halmos, P. R., A Hilbert Space Problem Book (1973), Springer: Springer New York, MR 84e:47001 · Zbl 0144.38704 [9] Han, J. K.; Lee, H. Y.; Lee, W. Y., Invertible completions of \(2 \times 2\) upper triangular operator matrices, Proc. Amer. Math. Soc., 128, 119-123 (2000), MR 2000c:47003 · Zbl 0944.47004 [10] Lee, W. Y., Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129, 131-138 (2001) · Zbl 0965.47011 [11] Rakovcevic, V., Approximate point spectrum and commuting compact perturbations, Glasg. Math. J., 28, 193-198 (1986) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.