Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A1⋯An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1⩽j⩽n, where A1,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rakočević/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rakočević/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rakočević/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.