The Kreiss Matrix Theorem on a General Complex Domain

Let A be a bounded linear operator in a Hilbert space $\H$ with spectrum $\Lam(A)$. The Kreiss matrix theorem gives bounds based on the resolvent norm $\norm{(zI-A)^{-1}}$ for $\norm{A^n}$ if $\Lam(A)$ is in the unit disk or for $\norm{e^{tA}}$ if $\Lam(A)$ is in the left half-plane. We generalize these results to a complex domain $\Ome$, giving bounds for $\norm{F_n(A)}$ if $\Lam(A) \subset \Ome$, where F_n denotes the nth Faber polynomial associated with $\Ome$. One of our bounds takes the form \til{\K}(\Ome) \; \leq \; 2\,\sup_{n} \, \norm{F_n(A)}, \qquad \norm{F_n(A)} \; \leq \; 2\,e \, (n+1) \, \til{\K}(\Ome), where $\til{\K}(\Ome)$ is the "Kreiss constant" defined by \til{\K}(\Ome) & = & \inf \left\{\, C \, : \, \norm{(zI-A)^{-1}} \; \leq \; C/\dist(z,\Ome) \ \forall \ z \not\in \Ome \right\}. 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, ${\rm dim}(\H) = N$, analogous results are also established in which ||F_n(A)|| is bounded in terms of N instead of n when the boundary of $\Ome$ is twice continuously differentiable.