Essential spectra, matrix operator and applications (Q983689)

In the two introductory sections of this paper, general notions of operator theory are discussed (Fredholm operators, the various notions of spectra, etc.). Then operators of the form \({A\;B\choose C\;D}\) acting in a Banach space \(X\times Y\) are considered; the operators \(A, B, C, D\) are not assumed to be bounded. A criterion is formulated under which the operator admits a closure \({\mathcal A}\). Then various types of spectra for \({\mathcal A}\) are indicated. Finally, applications are considered (delay differential equations, the Sturm-Liouville problem with the spectral parameter entering rationally, the transport equation). The exposition is not always accurate. For example, the authors write \(\lambda-T\), where \(T\) acts from the given space to another, the most important notion \(\rho_b(A_1)\) is explained too late, etc.