Operator inequalities implying similarity to a contraction

Abstract: Let $T$ be a bounded linear operator on a Hilbert space $H$ such that [ \alpha[T^*,T]:=\sum_{n=0}^\infty \alpha_n T^{*n}T^n\ge 0. ] where $\alpha(t)=\sum_{n=0}^\infty \alpha_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $\alpha(t) = (1-t) \tilde{\alpha} (t)$, where $\tilde{\alpha}$ has no roots in $[0,1]$, then $T$ is similar to a contraction. Operators of this type have been investigated by Agler, M\"uller, 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 $\alpha$ is imposed. Our approach is based on a factorization lemma for certain weighted $\ell^1$ Banach algebras.