  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.
