Fredholm properties of evolution semigroups

Abstract

We show that the Fredholm spectrum of an evolution semigroup $\{E^t\}_{t\geq 0}$ is equal to its spectrum, and prove that the ranges of the operator $E^t-I$ and the generator ${\bf G}$ of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banach space, and is induced by an evolution family that could be the propagator for a well-posed linear differential equation $u'(t)=A(t)u(t)$ with, generally, unbounded operators $A(t)$; in this case ${\bf G}$ is the closure of the operator $G$ given by $(Gu)(t)=-u'(t)+A(t)u(t)$.