Spectral value sets of closed linear operators
Eduardo Gallestey, D. Hinrichsen, A. J. Pritchard · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2000
We study how the spectrum of a closed linear operator on a complex Banach space changes under affine perturbations of the form A ↝ AΔ = A + DΔE. Here A, D and E are given linear operators, whereas Δ is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation DΔE. The union of the spectra of the perturbed operators AΔ, with the norm of Δ smaller than a given δ > 0, is called the spectral value set of A at level δ. 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 will be illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition.