As part of the theory of non-normal operators, the Kreiss matrix theorem states that if $A$ is a $N \times N$ matrix, and $\|\cdot\|$ is the spectral norm, then $$\mathcal{K}(A) \leq \sup_{k \geq 0} \|A^k\| \leq e N \mathcal{K}(A),$$ where $e$ is the Euler number and $\mathcal{K}(A)$ is the Kreiss constant of the matrix, defined by $$\mathcal{K}(A) = \sup_{z \in \mathbb{C} , |z|>1}(|z|-1)\|(zI - A)^{-1}\|$$
Does this theorem only hold for the spectral norm, or does it extend to any induced norm, i.e. $\|\cdot\|_p, p \in [1, \infty)$, at the price of perhaps changing the definition of $\mathcal{K}(A)$ to be in terms of this $\|\cdot\|_p$ norm? I could not find any reference for a more general claim. Any pointer would be much appreciated.
Note: It is immediately possible to derive naive upper and lower bounds using Hölder-type inequalities for matrices, but this breaks the dependency in $N$ of the Kreiss matrix theorem.