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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomials of generators of integrated semigroups


Author: Ralph deLaubenfels
Journal: Proc. Amer. Math. Soc. 107 (1989), 197-204
MSC: Primary 47D05; Secondary 47A60
MathSciNet review: 975637
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Abstract: We give general sufficient conditions on $ p$ and $ A$, for $ p(A)$ to generate an exponentially bounded holomorphic $ k$-times integrated semigroup, where $ p$ is a polynomial and $ A$ is a linear operator on a Banach space. Corollaries include the following.

(1) If $ iA$ generates a strongly continuous group and $ p$ is a polynomial of even degree with positive leading coefficient, then $ - p(A)$ generates a strongly continuous holomorphic semigroup of angle $ \frac{\pi } {2}$. (2) If $ - A$ generates a strongly continuous holomorphic semigroup of angle $ \theta $ and $ p$ is an $ n$th degree polynomial with positive leading coefficient, with $ n\left( {\tfrac{\pi } {2} - \theta } \right) < \tfrac{\pi } {2}$, then $ - p(A)$ generates a strongly continuous holomorphic semigroup of angle $ \tfrac{\pi } {2} - n(\tfrac{\pi } {2} - \theta )$. (3) If $ ( - A)$ generates an exponentially bounded holomorphic $ k$-times integrated semigroup of angle $ \theta $, and $ p$ and $ \theta $ are as in (2), then $ - p(A)$ generates an exponentially bounded holomorphic $ (k + 1)$-times integrated semigroup of angle $ \tfrac{\pi } {2} - n(\tfrac{\pi } {2} - \theta )$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0975637-7
PII: S 0002-9939(1989)0975637-7
Article copyright: © Copyright 1989 American Mathematical Society