Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Given the infinitesimal generator A of a C0-semigroup on the Banach space X which satisfies the Kreiss resolvent condition, i.e., there exists an M > 0 such that ‖(sI−A)‖ ≤ M 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 C0-semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow as much like t. Furthermore, we show that for every γ ∈ (0, 1) there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows like t . As a consequence, we find that for R with the standard Euclidian norm, the estimate ‖ exp(At)‖ ≤ M1 min(N, t) cannot be replaced by a lower power of N or t.