# Journal of Operator Theory

Volume 80, Issue 2,  Fall  2018  pp. 453-480.

Harnack parts of $\rho$-contractions

Authors:  Gilles Cassier (1), Mohammed Benharrat (2), and Soumia Belmouhoub (3)
Author institution: (1) Universite de Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 Novembre 1918, F-69622 Villeurbanne, France
(2) Departement de Mathematiques et informatique, Ecole Nationale Polytechnique d'Oran (Ex. ENSET d'Oran), B.P. 1523 Oran-El M'Naouar, Oran, Algerie
(3) Departement de Mathematiques, Universite de Mostaganem, Algerie

Summary:  The purpose of this paper is to describe the Harnack parts for the operators of class $C_{\rho}$ ($\rho>0$) on Hilbert spaces which were introduced by B.~Sz.-Nagy and C. Foias. 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 earlier. We obtain a general criterion for compact $\rho$-contractions to be in the same Harnack part. For classical contractions, this criterion can be simplified into a very useful form. Operators with numerical radius one receive also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

DOI: http://dx.doi.org/10.7900/jot.2017oct31.2174
Keywords:  $\rho$-contractions, Harnack parts, operator kernel, compact operators, operator radii, numerical range