Continuous-time kreiss resolvent condition on infinite-dimensional spaces

摘要

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 parallel to(sI - A)(-1)parallel to <= 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 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 epsilon (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 R-N with the standard Euclidian norm the estimate parallel to exp(At) parallel to <= M-1 min(N, t) cannot be replaced by a lower power of N or t.