Classes of contractions and Harnack domination

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.