## Illinois Journal of Mathematics

### 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 $(Gu)(t)=-u'(t)+A(t)u(t)$.

#### Article information

Source
Illinois J. Math., Volume 48, Number 3 (2004), 999-1020.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258131066

Digital Object Identifier
doi:10.1215/ijm/1258131066

Mathematical Reviews number (MathSciNet)
MR2114265

Zentralblatt MATH identifier
1073.34067

#### Citation

Latushkin, Yuri; Tomilov, Yuri. Fredholm properties of evolution semigroups. Illinois J. Math. 48 (2004), no. 3, 999--1020. doi:10.1215/ijm/1258131066. https://projecteuclid.org/euclid.ijm/1258131066