A new proof and generalizations of Gearhart’s theorem

Abstract:

Let $H$ be a Hilbert space, let $AP(\textbf {R},H)$ be the space of almost periodic functions from $\textbf {R}$ to $H$, and let $A$ be a closed densely defined linear operator on $H$. For a closed subset $\Lambda \subset \textbf {R}$, let $M(\Lambda )$ be the subspace of $AP(\textbf {R},H)$ 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 $\sup _{\lambda \in \Lambda }\|(i\lambda -A)^{-1}\|<\infty$. The case $\Lambda =\{2\pi k: k=0,\pm 1,\pm 2,...\}$ yields a new proof of the well-known Gearhart’s spectral mapping theorem.