Operator inequalities implying similarity to a contraction

Abstract

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ α[T^*,T]:=\sum_{n=0}^\infty α_n T^{*n}T^n\ge 0. \] where $α(t)=\sum_{n=0}^\infty α_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $α(t) = (1-t) \tildeα (t)$, where $\tildeα$ has no roots in $[0,1]$, then $T$ is similar to a contraction. Operators of this type have been investigated by Agler, Müller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy-Foias type model of an operator in this class and discuss its usual consequences (completeness of eigenfunctions, similarity to a normal operator, etc.). We also show that the limits of $\|T^nh\|$ as $n\to\infty$, $h\in H$, do not exist in general, but do exist if an additional assumption on $α$ is imposed. Our approach is based on a factorization lemma for certain weighted $\ell^1$ Banach algebras.