Essential spectra of some matrix operators involving \(\gamma\)-relatively bounded entries and an application. (English) Zbl 1502.47005
Summary: In this paper, we introduce the concept of relative boundedness with respect to an axiomatic measure of noncompactness \(\gamma\), as a generalization of the notion of relative boundedness. Then, some spectral properties of a \(2\times2\) unbounded block operator matrix involving \(\gamma\)-relatively bounded inputs are given. The obtained results are applied to investigate the essential spectra of a two-dimensional transport operator in \(L_1[(-a,a)\times(-1,1);dx\,dv]\) (\(0<a<\infty\)) with abstract boundary conditions relating the incoming and the outgoing fluxes.
MSC:
| 47A08 | Operator matrices |
| 47A10 | Spectrum, resolvent |
| 39B42 | Matrix and operator functional equations |
| 47A55 | Perturbation theory of linear operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
Keywords:
Fredholm theory; relatively bounded operator; measure of noncompactness; essential spectrumReferences:
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