Robust Stability ofC0-Semigroups and an Application to Stability of Delay Equations

AbstractLetAbe a closed linear operator on a complex Banach spaceXand let λ∈ϱ(A) be a fixed element of the resolvent set ofA. LetUandYbe Banach spaces, and letD∈L(U,X) andE∈L(X,Y) be bounded linear operators. We definerλ(A;D,E) by[formula]and prove that[formula]We give two applications of this result. The first is an exact formula for the so-called stability radius of the generator of aC0-semigroup of linear operators on a Hilbert space; it is derived from a precise result about robustness under perturbations of uniform boundedness in the right half-plane of the resolvent of an arbitrary semigroup generator. The second application gives sufficient conditions on the norm of the operatorsBj∈L(X) such that the classical solutions of the delay equation[formula]are exponentially stable inLp([−h,0];X)