This is the abstract of the paper
  "Almost Periodic Solutions of First and Second Order Cauchy Problems"
by W. Arendt and C.J.K. Batty.

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Abstract: Almost periodicity of solutions of first and
second order Cauchy problems on the real line is proved under the
assumption that the imaginary (resp. real) spectrum of the underlying
operator is countable. Related results have been obtained by Ruess-V\~{u}
and Basit. Our proof uses a new idea. It is based on a
factorisation method which also gives a short proof (of the
vector-valued version) of Loomis' classical theorem, saying that a bounded
uniformly continuous function from R into a Banach space $X$ with
countable spectrum is almost periodic if $c_0 \not\subset X$.
Our method can also be used for solutions on the half-line. This is done
in a separate paper.

To appear in Taiwanese J. Math.: Discrete Spectrum and almost periodicity