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

DOI:
https://doi.org/10.1090/S0002-9939-07-08691-1

Published electronically:
February 2, 2007

MathSciNet review:
2299482

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Hilbert space, let be the space of almost periodic functions from to , and let be a closed densely defined linear operator on . For a closed subset , let be the subspace of consisting of functions with spectrum contained in . We prove that the following properties are equivalent: (i) for every function there exists a unique mild solution of equation ; (ii) and . The case yields a new proof of the well-known Gearhart's spectral mapping theorem.

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Additional Information

**Vu Quoc Phong**

Affiliation:
Department of Mathematics, Ohio University, Athens, Ohio 45701

Email:
qvu@math.ohiou.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-08691-1

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.