Tauberian type theorem for operators with interpolation spectrum for Hölder classes

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.