# On the Spectrum of $C_0$-Semigroups

Jan Prüss
Transactions of the American Mathematical Society
Vol. 284, No. 2 (Aug., 1984), pp. 847-857
DOI: 10.2307/1999112
Stable URL: http://www.jstor.org/stable/1999112
Page Count: 11

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## Abstract

In this paper we give characterizations of the spectrum of a $C_0$- semigroup $e^At$ in terms of certain solution properties of the differential equation $(\ast) u' = Au + f$ and, in case $X$ is a Hilbert space, also in terms of properties of $(\lambda - A)^{-1}$. We give several applications of these results including a study of the existence of dichotomic projections for $(\ast)$.

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