Acta Mathematica Vietnamica

We study properties of solutions of operator equations $\mathcal Du−\mathcal Bu = f\,\, (∗)$ where $\mathcal D$ is the generator of an isometric group $V(t)$ on a Banach space $\mathcal F$ and $\mathcal B$ is a closed operator commuting with $\mathcal D$. We introduce the equation spectrum $\Sigma$ and prove that if $f$ is an almost periodic element (with respect to the group $V(t)$) and $\Sigma$ is countable, then any solution $u$ of $(*)$ is almost periodic, provided either $\mathcal F \not\supset c_0$ or $u$ is totally ergodic. The presented approach when applied to functional-differential equations gives spectral criteria of almost periodicity of bounded uniformly continuous solutions. The discrete version of the results, with applications to properties of solutions of functionaldifference equations, also is described.

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