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We present a general spectral decomposition technique for bounded solutions toinhomogeneous linear periodic evolution equations of the form  x = A(t)x+f(t) (),with f having precompact range, which will be then applied to find new spectralcriteria for the existence of almost periodic solutions with specific spectral propertiesin the resonnant case where eisp(f)may intersect the spectrum of the monodromyoperator P of () (here sp(f) denotes the Carleman spectrum of f ). We show that...

New spectral criteria for almost periodic solutions of evolution equations