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A new proof and generalizations of Gearhart's theorem

Author: Vu Quoc Phong
Journal: Proc. Amer. Math. Soc. 135 (2007), 2065-2072
MSC (2000): Primary 47D06, 35B40
Published electronically: February 2, 2007
MathSciNet review: 2299482
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Abstract: 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.

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Vu Quoc Phong
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701

Keywords: $C_0$-semigroup, almost periodic, admissible subspace, spectral mapping theorem
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.