A generalization of the Kreiss Matrix Theorem

Let Ω be a compact subset of the complex plane such that its complement is simply connected in the extended complex plane. Suppose A is a linear bounded operator in a Hilbert space, with spectrum σ(A)⊂Ω. If Ω is symmetric with respect to the real line and f is a Markov function, we show that‖f(A)‖≤eCK(Ω)‖f‖Ω, where K(Ω) is the Kreiss constant with respect to Ω and C is a constant. We also present other extensions of the Kreiss Matrix Theorem to arbitrary holomorphic functions.