The Kreiss matrix theorem on a general complex domain

Let A be a bounded linear operator in a Hilbert space H with spectrum Lambda(A). The Kreiss matrix theorem gives bounds based on the resolvent norm parallel to(zI - A)(-1) parallel to for parallel to A(n)parallel to if Lambda(A) is in the unit disk or for parallel to e(tA)parallel to if Lambda(A) is in the left half-plane. We generalize these results to a complex domain Omega, giving bounds for parallel to F-n(A)parallel to if Lambda(A) subset of Omega, where F-n denotes the nth Faber polynomial associated with Omega. One of our bounds takes the form (K) over tilde(Omega) less than or equal to 2 sup parallel to F-n(A)parallel to, parallel to Fn(A)parallel to less than or equal to 2e(n + 1)(K) over tilde(Omega), where (K) over tilde(Omega) is the "Kreiss constant" defined by (K) over tilde(Omega) = inf {C:parallel to(zI - A)(-1)parallel to less than or equal to C/dist(z, Omega) For All z is not an element of Omega}. By means of an inequality due originally to Bernstein, the second inequality can be extended to general polynomials p(n). In the case where H is finite-dimensional, say, dim(H) = N, analogous results are also established in which parallel to F-n(A)parallel to is bounded in terms of N instead of n when the boundary of Omega is twice continuously differentiable.