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Q.-P. Vu (Ohio University)

Title: A spectral theory of abstract evolution equations and its applications

Abstract: Let $D$ be a generator of an isometric group $V(t)$ on a Banach space $E$ and $B$ be a closed operator on $E$ which commutes with $V(t)$. We present a spectral theory of the operator equation $Du-Bu=f$, and apply it to obtain conditions for almost periodicity of solutions of abstract evolution equations $u'(t)=({\cal B}u) +f(t)$.
We also consider the case when $D$ is the generator of a bounded $C_0$-semigroup $S(t), t\ge 0$, and obtain conditions for asymptotic almost periodicity of solutions of $Du-Bu=f$, where $f$ is asymptotically almost periodic (w.r.t. the semigroup $S(t)$), obtaining in this way a generalization of results obtained jointly with Yu. I. Lyubich on spectral conditions of almost periodicity of $C_0$-semigroups.
A related question of regular admissibility of a subspace $M$ of $E$ is also considered. For the case of equations on Hilbert space the results are generalizations of Gearhart's Theorem.