HARNACK parts of $ρ$-Contractions

The purpose of this paper is to describe the Harnack parts for the operators of class C $ρ$ ($ρ$ \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 $ρ$-numerical radius one. The case of operators with $ρ$-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact $ρ$-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.

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