HARNACK parts of $\rho$-Contractions

TL;DR

This paper characterizes the Harnack parts of operators with numerical radius one in the class C rho on Hilbert spaces, extending previous work and providing criteria for compact rho-contractions to be Harnack equivalent.

Contribution

It introduces a general criterion for compact rho-contractions to belong to the same Harnack part and explores properties of Harnack equivalence for these operators.

Findings

Criteria for compact rho-contractions in the same Harnack part

Characterization of Harnack parts for operators with numerical radius one

Analysis of Harnack equivalence properties

Abstract

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.