Operator inequalities implying similarity to a contraction

TL;DR

This paper establishes conditions under which a bounded linear operator satisfying a specific operator inequality is similar to a contraction, extending previous results with explicit models and new techniques.

Contribution

It proves that operators satisfying a certain positivity condition are similar to contractions when a factorization condition on the defining function is met, providing explicit models and analysis.

Findings

Operators satisfying the inequality are similar to contractions under certain conditions.

An explicit Nagy-Foias type model is constructed for these operators.

Limits of the norms of iterates do not always exist without additional assumptions.

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.