A Generalization of the Kreiss Matrix Theorem

Let $\mathcal{A}$ be a set of $n \times n$ complex matrices A which satisfy the condition $\| {(I - zA)} \| \leqq {K / {(1 - | z |)}}^{\alpha + 1} $ for some $\alpha \geqq 0$ and all $| z | < 1$. Then it is shown here that there exists a constant $\rho (\alpha ,n)$ such that $\| {A^\nu } \| \leqq K_\rho (\alpha ,n)\nu ^\alpha $, $\nu = 0,1, \cdots $. This forms a generalization of the Kreiss resolvent condition (where $\alpha = 0$).