Harnack parts for some truncated shifts

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.