Chen, Xiaoling; Zhang, Shifang; Zhong, Huaijie On the filling in holes problem for operator matrices. (English) Zbl 1151.47005 Linear Algebra Appl. 430, No. 1, 558-563 (2009). Summary: We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that, in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces. Cited in 5 Documents MSC: 47A10 Spectrum, resolvent 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators Keywords:operator matrices; Browder essential approximate point spectrum; upper semi-Fredholm spectrum; lower semi-Fredholm spectrum PDFBibTeX XMLCite \textit{X. Chen} et al., Linear Algebra Appl. 430, No. 1, 558--563 (2009; Zbl 1151.47005) Full Text: DOI References: [1] Cao, X. H.; Guo, M. Z.; Meng, B., Weyl’s theorem for upper triangular operator matrices, Linear Algebra Appl., 402, 61-73 (2005) · Zbl 1129.47301 [2] Cao, X. H.; Guo, M. Z.; Meng, B., Semi-Fredholm spectrum and Weyl’s theory for operator matrices, Acta Math. Sin., 22, 169-178 (2006) · Zbl 1129.47014 [3] Cao, X. H.; Guo, M. Z.; Meng, B., Drazin spectrum and Weyl’s theorem for operator matrices, J. Math. Res. Exposition, 26, 413-422 (2006) · Zbl 1118.47004 [4] Cao, X. H., Browder spectra for upper triangular operator matrices, J. Math. Anal. Appl., 342, 477-484 (2008) · Zbl 1139.47006 [5] Djordjević, D. S., Perturbations of spectra of operator matrices, J. Operator Theory, 48, 467-486 (2002) · Zbl 1019.47003 [6] Djordjević, S. V.; Han, Y. M., A note on Weyl’s theorem for operator matrices, Proc. Amer. Math. Soc., 131, 2543-2547 (2002) · Zbl 1041.47006 [7] Djordjević, S. V.; Han, Y. M., Browder’s theorem and spectral continuity, Glasgow Math. J., 42, 479-486 (2000) · Zbl 0979.47004 [8] Du, H. K.; Pan, J., Perturbation of spectrums of \(2 \times 2\) operator matrices, Proc. Amer. Math. Soc., 121, 761-766 (1994) · Zbl 0814.47016 [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 (1999) [10] 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 [11] Lee, W. Y., Weyl’s theorem for operator matrices, Integral Equations Operator Theory, 32, 319-331 (1998) · Zbl 0923.47001 [12] Lee, W. Y., Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129, 131-138 (2000) · Zbl 0965.47011 [13] Zhang, Shifang; Zhong, Huaijie, A note of Browder spectrum of operator matrices, J. Math. Anal. Appl., 344, 927-931 (2008) · Zbl 1146.47004 [14] Zhang, Shifang; Zhong, Huaijie; Jiang, Qiaofen, Drazin spectrum of operator matrices on the Banach space, Linear Algebra Appl., 429, 2067-2075 (2008) · Zbl 1157.47004 [15] Zhang, Yunnan; Zhong, Huaijie; Lin, Liqiong, Browder spectra and Essential Spectra of Operator Matrices, Acta Math. Sin., 24, 947-954 (2008) · Zbl 1162.47004 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.