C-semigroups and strongly continuous semigroups |
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Authors: | Ralph deLaubenfels |
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Institution: | (1) Department of Mathematics, Ohio University, 45701 Athens, OH, USA |
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Abstract: | We show that, whenA generates aC-semigroup, then there existsY such that M(C)] →Y →X, andA|
Y
, the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖Im(C)]≡‖C
−1
x‖. There also existsW such that C(W)] →X →W, and an operatorB such thatA=B|
X
andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true.
We construct fractional powers of generators of boundedC-semigroups.
We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference 37]. This research was
supported by an Ohio University Research Grant. |
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Keywords: | |
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