Li, Yuan; Sun, Xiu-Hong; Du, Hong-Ke The intersection of left (right) spectra of \(2 \times 2\) upper triangular operator matrices. (English) Zbl 1105.47004 Linear Algebra Appl. 418, No. 1, 112-121 (2006). Let \(H, K\) be two Hilbert spaces, \(B(H)\) and \(B(H,K)\) be the set of bounded operators on \(H\) and from \(H\) to \(K\), respectively. In the paper under review, the sets \(\bigcup_{C\in B(H, K)}\sigma_l(M_C)\) and \(\bigcup_{C\in \text{Inv}(H, K)}\sigma_l(M_C)\) are compared. Here \(\text{Inv}(H,K)\) stands for the set of invertible operators from \(H\) to \(K\), \(\sigma_l(.)\) denotes the left spectrum and \(M_C \in B(H\oplus K)\) an upper triangular operator matrix with \(A\in B(H), B\in B(K)\) the entries on the diagonal and \(C \in B(H,K)\) the remaining non-vanishing entry. The main result is \(\bigcup_{C\in B(H, K)}\sigma_l(M_C)= \bigcup_{C\in \text{Inv}(H, K)}\sigma_l(M_C)\cup \{\lambda \in \mathbb C : B-\lambda \: \text{ is compact}\}\). To get this result, one has to assume that both \(H\) and \(K\) are infinite-dimensional (otherwise the right term is always the whole complex plane and the left one is a subset of \(\sigma(A)\cup \sigma(B)\)). Moreover, in the infinite-dimensional case, it is not difficult to see that \(\{\lambda \in \mathbb C : B-\lambda \text{ is compact}\}\) is at most a singleton. Reviewer: El Hassan Zerouali (Rabat) Cited in 17 Documents MSC: 47A10 Spectrum, resolvent 47A55 Perturbation theory of linear operators Keywords:operator matrices; perturbation of spectra PDFBibTeX XMLCite \textit{Y. Li} et al., Linear Algebra Appl. 418, No. 1, 112--121 (2006; Zbl 1105.47004) Full Text: DOI References: [1] Conway, J. B., A Course in Functional Analysis (1989), Springer: Springer New York [2] Djordjevic, D. S., Perturbations of spectra of operator matrices, J. Operator Theory., 48, 467-486 (2002) · Zbl 1019.47003 [3] Djordjevic, S. V.; Han, Y. M., Browder’s theorems and Spectral continuity, Glasgow. Math. J., 42, 479-486 (2000) · Zbl 0979.47004 [4] Du, H. K.; Pan, J., Perturbations of spectra of 2×2 operator matrices, Proc. Amer. Math. Soc., 121, 761-767 (1994), MR 94i:47004 [5] Fillmore, P. A.; Willams, J. P., On operator ranges, Adv. Math., 7, 254-281 (1971) · Zbl 0224.47009 [6] Hwang, I. S.; Lee, W. Y., The boundedness below of 2×2 upper triangular operator matrices, Integr. Equat. Oper. Th., 39, 267-276 (2001) · Zbl 0986.47004 [7] Halmos, P. R., A Hilbert Space Problem Book (1973), Springer: Springer New York, MR 84e:47001 · Zbl 0144.38704 [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), MR 2000c:47003 · Zbl 0944.47004 [9] Lee, W. Y., Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129, 131-138 (2001) · Zbl 0965.47011 [10] Li, Yuan; Sun, Xiuhong; Du, Hongke, The intersection of left (right) Weyl spectra of 2×2 upper triangular operator matrices, Acta Math. Sin., 48, 653-660 (2005), (in Chinese) · Zbl 1125.47301 [11] Yuan Li, Hongke Du, The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., in press.; Yuan Li, Hongke Du, The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., in press. · Zbl 1108.47013 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.