IUMJ

Title: Weakly wandering vectors and interpolation theorems for power bounded operators

Authors: Vladimir Mueller and Yuri Tomilov

Issue: Volume 59 (2010), Issue 3, 1121-1144

Abstract:

Let $\mathbf{c} = \{c_{p,q}\} \subset \mathbb{C}$, $p$, $q \in \mathbb{N}$, $p > q$, be such that $\mathrm{D}$-$\lim_{p \to \infty} c_{p,q} = 0$ for each $q \in \mathbb{N}$, and let $T$ be a power bounded operator on a Hilbert space $H$ with infinite peripheral spectrum and empty point peripheral spectrum. We prove that $\mathbf{c}$ can be interpolated by the orbits of $T$ in the sense that the set of $x$'s from $H$ with $\langle T^{n_k}x,T^{n_{k'}}x\rangle=c_{n_k, n_{k'}}$ for a certain increasing sequence $\{n_k\} \subset \mathbb{N}$ (depending on $x$) and all $k$, $k' \in \mathbb{N}$, $k > k'$, is dense in $H$. In particular, the set of weakly wandering vectors for such $T$ is dense in $H$. This extends previous similar results known only in the context of unitary representations of groups. Our results are optimal as far as spectral conditions are concerned. Moreover, our technique allows one to treat operators whose sequence of powers is unbounded.