Generalized eigenspaces of generators of evolution semigroups

A periodic evolutionary process {U(t,s)}t≥s{U(t,s)}t≥s with period τ on a Banach space X gives an evolution semigroup with the generator L on the space of τ-periodic continuous X  -valued functions. We prove that the generalized eigenspaces NL=N((αI−L)n)NL=N((αI−L)n) and NU=N((eατI−U(τ,0))n)NU=N((eατI−U(τ,0))n) are isomorphic. The algebraic representation of a function u∈NLu∈NL is given by the initial value u(0)=w∈NUu(0)=w∈NU.