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On the filling in holes problem for operator matrices. (English) Zbl 1151.47005
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.
##### MSC:
 47A10 Spectrum and resolvent of linear operators 47A53 (Semi-)Fredholm operators; index theories 47A55 Perturbation theory of linear operators
##### References:
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