Tauberian type theorem for operators with interpolation spectrum for Hölder classes

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 $\|T^{n}\|=O(n^p)$, $p\geq 1$, $\|T^{-n}\|=O(w_n)$ 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 $\|T^{-n}\|=O(n^{p+s})$ 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.