Two counterexamples related to the Kreiss matrix theorem (Q1343045)

Let \(A\) be an \(s\times s\) matrix satisfying certain resolvent condition with respect to the unit disk with constant \(K\). A recent version of the Kreiss matrix theorem states that \(|A^n |\leq esK\), \(n\geq 0\). In this paper the author shows that for any fix \(K\geq \pi+1\) the upper bound \(esK\) is sharp in the sense that a linear dependence on the dimension \(s\) is the best possible result. Also, a similar result is obtained for the continuous version, which states that \(|e^{tA} |\leq esK\), \(t>0\), if \(A\) satisfies a resolvent condition with respect to the left half plane with constant \(K\).