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Generalized invertibility of operator matrices. (English) Zbl 1264.47001

Let Z be a Banach space, such that Z=XY for some closed and complementary subspaces X and Y. Then each operator M𝔹(Z) which is invariant on X, has a decomposition into some operators A𝔹(X), B𝔹(Y) and C𝔹(Y,X).

In this paper, the authors discuss under which conditions on A and B, there exists some C such that M has a generalized inverse, or is left Browder (that is, left Fredholm with finite ascent), or has some other similar properties.

MSC:
47A05General theory of linear operators
47A10Spectrum and resolvent of linear operators
47A53(Semi-)Fredholm operators; index theories
References:
[1]Cao, X., Browder essential approximate point spectra and hypercyclicity for operator matrices, Linear Algebra Appl. 426 (2007), 317–324. · Zbl 1133.47004 · doi:10.1016/j.laa.2007.05.003
[2]Djordjević, D. S., Perturbations of spectra of operator matrices, J. Operator Theory 48 (2002), 467–486.
[3]Hai, G. and Chen, A., Perturbations of right and left spectra for operator matrices, to appear in J. Operator Theory.
[4]Han, J. K., Lee, H. Y. and Lee, W. Y., Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000), 119–123. · Zbl 0944.47004 · doi:10.1090/S0002-9939-99-04965-5
[5]Han, Y. M. and Djordjević, S. V., a-Weyl’s theorem for operator matrices, Proc. Amer. Math. Soc. 130 (2001), 715–722. · Zbl 0994.47013 · doi:10.1090/S0002-9939-01-06110-X
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[8]Zhang, S., Wu, Z. and Zhong, H., Continuous spectrum, point spectrum and residual spectrum of operator matrices, Linear Algebra Appl. 433 (2010), 653–661. · Zbl 1197.47013 · doi:10.1016/j.laa.2010.03.036