## A new proof and generalizations of Gearhart’s theorem

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**135**(2007), 2065-2072 Request permission

## 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.## References

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## Additional Information

**Vu Quoc Phong**- Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
- Email: qvu@math.ohiou.edu
- Received by editor(s): December 29, 2005
- Received by editor(s) in revised form: March 2, 2006
- Published electronically: February 2, 2007
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**135**(2007), 2065-2072 - MSC (2000): Primary 47D06, 35B40
- DOI: https://doi.org/10.1090/S0002-9939-07-08691-1
- MathSciNet review: 2299482