HARNACK parts of $\rho$-Contractions

The purpose of this paper is to describe the Harnack parts for the operators of class C $\rho$ ($\rho$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with $\rho$-numerical radius one. The case of operators with $\rho$-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact $\rho$-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.