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Mathematics of Computation

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Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Authors: Tatjana Eisner and Hans Zwart
Journal: Math. Comp. 75 (2006), 1971-1985
MSC (2000): Primary 47D06, 15A60; Secondary 65J10, 34K20, 47N40
Published electronically: July 10, 2006
MathSciNet review: 2240644
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Abstract: Given the infinitesimal generator $ A$ of a $ C_0$-semigroup on the Banach space $ X$ which satisfies the Kreiss resolvent condition, i.e., there exists an $ M>0$ such that $ \Vert (sI-A)^{-1}\Vert \leq \frac{M}{\mathrm {Re}(s)}$ for all complex $ s$ with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated $ C_0$-semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like $ t$. Furthermore, we show that for every $ \gamma \in (0,1)$ there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like $ t^\gamma$. As a consequence, we find that for $ {\mathbb{R}}^N$ with the standard Euclidian norm the estimate $ \Vert\exp(At)\Vert \leq M_1 \min(N,t)$ cannot be replaced by a lower power of $ N$ or $ t$.

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Additional Information

Tatjana Eisner
Affiliation: Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Hans Zwart
Affiliation: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Keywords: Kreiss resolvent estimate, $C_0$-semigroups, stability estimate
Received by editor(s): March 14, 2005
Received by editor(s) in revised form: September 13, 2005
Published electronically: July 10, 2006
Dedicated: Dedicated to M.N. Spijker on the occasion of his 65th birthday.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.