Relatively strictly singular perturbations, essential spectra, and application to transport operators (Q1840541)

If the operator \(A\) we think of is a densely defined closed linear operator in a Banach space \(X\), there are several definitions of the essential spectrum \(\sigma_e(A)\) of \(A\), for instance, in the sense of Gustafson-Weidmann, Kato, Wolf, Schechter, Browder and so on. They are closed sets of \(\lambda\) in the complex plane \(\mathbb{C}\), which are closed subsets of the spectrum \(\sigma(A)\) of \(A\) defined with \(\text{nul}(A- \lambda)\), \(\text{def}(A- \lambda)\), \(\text{ind}(A- \lambda)\) and some other concepts. All these essential spectra coincide if \(A\) is a selfadjoint operator in a Hilbert space. As a continuation of a previous paper by the first author and \textit{A. Jeribi} [J. Math. Anal. Appl. 225, No. 2, 461-485 (1998; Zbl 0927.47007)], the paper under review studies further, when \(A\) is a densely defined closed linear operator in special Banach spaces \(L^p\)-spaces, \(1\leq p\leq\infty\), with domain \(D(A)\), the invariance of the essential spectrum in the sense of Gustafson-Weidmann, Wolf, Schechter and Browder under relatively strictly singular perturbations. Here a linear operator \(S\) in \(X\) is said to be relatively strictly singular with respect to \(A\), if \(S\) is \(A\)-bounded and if, for every infinite-dimensional subspace \(M\) of \(D(A)\) as a Banach space equipped with the graph norm of \(A\), the restriction of \(S\) to \(M\) is not a homeomorphism of \(D(A)\) into \(X\). So a relatively strictly singular operator is not necessarily bounded. In the latter half of the paper, the result is applied to describe the essential spectra of the one-dimensional transport equations with anisotropic scattering and abstract boundary conditions.