Almost periodic at infinity functions from homogeneous spaces as solutions to differential equations with unbounded operator coefficients

By using the subspace of functions from homogeneous spaces with integrals decreasing at infinity we define new classes of functions almost periodic at infinity. We obtain spectral criteria for a function to be almost periodic at infinity and asymptotically almost periodic (with respect to the chosen subspace). These results are used for deriving criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. In addition, for the new class of asymptotically finite-dimensional operator semigroups we prove the almost periodicity at infinity of their orbits.