A new proof and generalizations of Gearhart's theorem

Let $H$ be a Hilbert space, let $AP({\bf R},H)$ be the space of almost periodic functions from ${\bf R}$ to $H$, and let $A$ be a closed densely defined linear operator on $H$. For a closed subset $\Lambda\subset {\bf R}$, let $M(\Lambda)$ be the subspace of $AP({\bf R},H)$ consisting of functions with spectrum contained in $\Lambda$. We prove that the following properties are equivalent: (i) for every function $f\in M(\Lambda)$ there exists a unique mild solution $u\in M(\Lambda)$ of equation $u'(t)=Au(t)+f(t)$; (ii) $i\Lambda\subset \rho(A)$ and $\sup_{\lambda \in\Lambda}\Vert(i\lambda -A)^{-1}\Vert<\infty$. The case $\Lambda=\{2\pi k: k=0,\pm1,\pm2,...\}$ yields a new proof of the well-known Gearhart's spectral mapping theorem.