Harnack parts for some truncated shifts

TL;DR

This paper analyzes the Harnack parts of certain finite-dimensional truncated shifts with a specific numerical radius, focusing on null spaces of associated operators and providing new fundamental results.

Contribution

It introduces new descriptions of null spaces for the $ ho$-operatorial kernel of truncated shifts, advancing understanding of their Harnack parts.

Findings

Characterization of null spaces of $ ho$-operatorial kernels

Two fundamental results on Harnack parts of truncated shifts

Applications to finite-dimensional operator analysis

Abstract

The purpose of this paper is to analysis the Harnack part of some truncated shifts whose $\rho$-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the $\rho$-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given.