Spectral value sets of closed linear operators (Q2713408)

From authors' abstract: We study how the spectrum of a closed linear operator on a complex Banach space changes under affine perturbation of the form \(A\to A_\Delta= A+ D\Delta E\). Here \(D\), \(A\), and \(E\) are given linear operators, whereas \(\Delta\) is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation \(D\Delta E\). The union of the spectra of the perturbed operators \(A_\Delta\), with the norm of \(\Delta\) smaller than a given \(\delta> 0\), is called the spectral value set of \(A\) at level \(\delta\). In this paper we extend a known characterization of these sets for the matrix case to infinite dimensions, and in so doing present a framework that allows for unbounded perturbations of closed linear operators on Banach spaces. The results are illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition.