Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Eisner, T. and Zwart, H.J. (2006) Continuous-time Kreiss resolvent condition on infinite-dimensional spaces. Mathematics of Computation, 75 (256). pp. 1971-1985. ISSN 0025-5718

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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 $t$ .
Item Type:	Article
Faculty:	Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:	Mathematical Systems and Control Theory (MSCT)
Link to this item:	http://purl.utwente.nl/publications/63837
Official URL:	http://www.ams.org/mcom/2006-75-256/S0025-5718-06-01862-X/home.html
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