Abstract: We consider an invertible operator $ T$ on a Banach space $ X$ whose spectrum is an interpolating set for Hölder classes. We show that if $ \Vert T^{n}\Vert=O(n^p)$, $ p\geq1$, $ \Vert T^{-n}\Vert=O(w_n)$ with $ n^q=o(w_n)$ $ \forall q\in\mathbb{N}$ and $ \sum_n 1/(n^{1-\alpha} (\log w_{n})^{1+\alpha})=+\infty$, then $ \Vert T^{-n}\Vert=O(n^{p+s})$ for all $ s > \tfrac{1}{2}$, assuming that $ (w_n)_{n\geq 1}$ satisfies suitable regularity conditions. When $ X$ is a Hilbert space and $ p=0$ (i.e. $ T$ is a contraction), we show that under the same assumptions, $ T$ is unitary and this is sharp.