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On characterization and perturbation of local $ C$-semigroups

Authors: Yuan-Chuan Li and Sen-Yen Shaw
Journal: Proc. Amer. Math. Soc. 135 (2007), 1097-1106
MSC (2000): Primary 47D06, 47D60
Published electronically: September 26, 2006
MathSciNet review: 2262911
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Abstract: Let $ S(\cdot)$ be a $ (C_0)$-group with generator $ -B$, and let $ \{T(t);0\le t<\tau\}$ be a local $ C$-semigroup commuting with $ S(\cdot)$. Then the operators $ V(t):=S(-t)T(t)$, $ 0\le t<\tau$, form a local $ C$-semigroup. It is proved that if $ C$ is injective and $ A$ is the generator of $ T(\cdot)$, then $ A+B$ is closable and $ \overline{A+B}$ is the generator of $ V(\cdot)$. Also proved are a characterization theorem for local $ C$-semigroups with $ C$ not necessarily injective and a theorem about solvability of the abstract inhomogeneous Cauchy problem: $ u'(t)=Au(t)+Cf(t), 0<t<\tau; u(0)=Cx.$

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Additional Information

Yuan-Chuan Li
Affiliation: Department of Applied Mathematics, National Chung-Hsing University, Taichung, 402 Taiwan

Sen-Yen Shaw
Affiliation: Graduate School of Engineering, Lunghwa University of Science and Technology, Gueishan, Taoyuan, 333 Taiwan

Keywords: Local $C$-semigroup, $(C_0)$-group, generator, perturbation
Received by editor(s): August 8, 2005
Received by editor(s) in revised form: November 7, 2005
Published electronically: September 26, 2006
Additional Notes: This research was supported in part by the National Science Council of Taiwan.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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