Perturbing Exponentially Dichotomous Operators

Abstract

In this chapter our ultimate goal is to prove (or disprove) that bounded additive perturbations S of exponentially dichotomous operators S₀ are exponentially dichotomous, provided there exists a vertical strip of the form

$$ \left\{ {\lambda \in \mathbb{C}:\left| {\operatorname{Re} \lambda } \right| \leqslant \varepsilon } \right\} $$

within the resolvent set of the perturbed operator S and the resolvent (λ-S)⁻¹ is bounded on this strip. Although at first sight the solution of this perturbation problem seems to be a piece of cake (as it obviously is in the semigroup case), the dependence of the separating projection of S on the perturbation considerably complicates the problem. Our basic strategy is to derive the perturbed bisemigroup bt solving a vector-valued convolution equation on the line using the unperturbed bisemigroup as an integral kernel and the unperturbed bisemigroup acting on an arbitrary vector as the inhomogeneous term. This requires representing pointwise inverses of Fourier transforms of operator-valued functions as Fourier transforms of operator-valued functions. We therefore first discuss basic Gelfand theory of commutative Banach algebras and derive the classical Allan-Bochner-Phillips theorem on inversion within the operator-valued Wiener algebra. We then develop additive perturbation theory if the perturbation is a compact operator or the bisemigroup is analytic (or at least immediately norm continuous). Here we can remain within the comfortable realm of Bochner integrals of vector-valued functions.