### 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$.

Original language | Undefined |
---|---|

Pages (from-to) | 1971-1985 |

Number of pages | 15 |

Journal | Mathematics of computation |

Volume | 75 |

Issue number | suppl 2/256 |

Publication status | Published - 10 Jul 2006 |

### Keywords

- EWI-8605
- MSC-47D06
- METIS-237798
- IR-63837
- MSC-15A60

## Cite this

Eisner, T., & Zwart, H. J. (2006). Continuous-time Kreiss resolvent condition on infinite-dimensional spaces.

*Mathematics of computation*,*75*(suppl 2/256), 1971-1985.