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
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