Classes of contractions and Harnack domination (Q529988)

Let \(T,T'\) be Hilbert space contractions, i.e., linear operators having norm less than or equal to one. \(T\) is Harnack dominated by \(T'\) (\(T\overset{\text{H}}\prec T'\)) if there exists \(c\geq 1\) such that \(\Re p(T)\leq \Re p(T')\) for any polynomial \(p(z)\) for which \(|z|\leq 1\) implies \(\Re p(z)\geq 0\). Relations between Harnack domination and several properties of the operators \(T\) and \(T'\) are studied in this paper. For example, it is shown that \(T\) is maximal (resp., minimal) with respect to relation \(\overset{\text{H}}\prec\) if and only if it is singular unitary (resp., isometry or coisometry). Other interesting results describe what happens if \(T\) and/or \(T'\) belong to one of the Sz.-Nagy Foiaş classes \(C_{0\cdot},C_{\cdot0}, C_{00}\). The paper also contains some examples of particular operators \(T,T'\) that illustrate the obtained results.