Classes of contractions and Harnack domination

TL;DR

This paper investigates the properties of Harnack domination among Hilbert space contractions, identifying extremal elements and showing how various operator properties transfer under this relation.

Contribution

It characterizes maximal and minimal elements in Harnack domination and demonstrates the transfer of key properties from contractions to Harnack dominated operators.

Findings

Maximal elements are singular unitary operators.

Minimal elements are isometries and their adjoints.

Properties like convergence and spectral features transfer under Harnack domination.

Abstract

Several properties of the Harnack domination of linear operators acting on Hilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g. convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it.