A periodic evolutionary process {U(t, s)}(t >= s), with period T on a Banach space X gives an evolution semigroup with the generator L on the space of T-periodic continuous X-valued functions. We prove that the generalized eigenspaces ATL = N((alpha I - L)(n)) and Nu = N((e(alpha tau) I - U (tau,0))(n)) are isomorphic. The algebraic representation of a function u is an element of N-L, is given by the initial value u(0) = w is an element of Nu. (C) 2015 Elsevier Inc. All rights reserved.