Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

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$.