Operator inequalities implying similarity to a contraction

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that $$\alpha [T^{\ast },T]:=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n{T}^{\ast n}{T}^n\ge 0.$$ where $\alpha (t)={\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n{t}^n$ 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$\tilde{\alpha}$hasnorootsin$[0,1]$, 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 $\mathrm{\parallel }{T}^{n}h\mathrm{\parallel }$ as $n\to \mathrm{\infty }$, $h\in H$, 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 ${\mathrm{\ell }}^1$ Banach algebras.