New spectral criteria for almost periodic solutions of evolution equations

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form (x) over dot = A(t)x + f(t) (*), with f having precompact range, which is then appli ed to find new spectral criteria for the existence of almost periodic solut ions with specific spectral properties in the resonant case where e(i sp(f) ) may intersect the spectrum of the monodromy operator P of (*) there sp(f) denotes the Carleman spectrum of (f). We show that if (*) has a bounded un iformly continuous mild solution u and sigma (Gamma)(P)\e(isp(f)) is closed , where sigma (Gamma)(P) denotes the part of sigma (P) on the unit circle, then (*) has a bounded uniformly continuous mild solution w such that e(i s p(w)) = e(i sp(f)). Moreover, w is a "spectral component" of u. This allows us to solve the general Massera-type problem for almost periodic solutions . Various spectral criteria for the existence of almost periodic and quasi- periodic mild solutions to (*) are given.