Two Counterexamples Related to the Kreiss Matrix Theorem

摘要

The modern version of the Kreiss matrix theorem states that matrix norm A(sup n)= or < esK (for all n = or > 0) if A is an s X s matrix satisfying a resolvent condition with respect to the unit disk with constant K. In this paper we show for any fixed K = or > pi + 1 that the upper bound esK is sharp in the sense that a linear dependence on the dimension s is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that Norm exp (tA) = or < esK (for all t = or > 0) if A is an s X s matrix satisfying a resolvent condition with respect to the left half plane with constant K.