On the spectrum of 𝐶₀-semigroups

Source abstract

In this paper we give characterizations of the spectrum of a C 0 {C_0} -semigroup e A t {e^{At}} in terms of certain solution properties of the differential equation ( ∗ ) u ′ = A u + f (\ast )\;u’ = Au + f and, in case X X is a Hilbert space, also in terms of properties of ( λ − A ) − 1 {(\lambda - A)^{ - 1}} . We give several applications of these results including a study of the existence of dichotomic projections for ( ∗ ) (\ast ) .