Illinois Journal of Mathematics

Fredholm properties of evolution semigroups

Yuri Latushkin and Yuri Tomilov

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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

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

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 35F10: Initial value problems for linear first-order equations 35P05: General topics in linear spectral theory 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]


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

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