Harnack parts for 5-by-5 truncated shift with numerical radius one

TL;DR

This paper characterizes the Harnack parts of a normalized 5x5 truncated shift with numerical radius one, revealing a richer algebraic structure than in lower dimensions.

Contribution

It provides a complete description of the Harnack component for the 5x5 truncated shift, identifying two distinct operator forms within this class.

Findings

Operators in the Harnack part are either unitarily equivalent to the shift or are structured nilpotent matrices.

All elements in the Harnack class are necessarily nilpotent with the same order as the truncated shift.

The algebraic structure of the Harnack class in dimension five is more complex than in lower dimensions.

Abstract

We provide a complete description of the Harnack part for normalized truncated shift of size five with numerical radius one. We prove that any operator in this Harnack component must assume one of two distinct forms: either it belongs to the unitary orbit of the shift, thereby preserving its norm, or it is a structured nilpotent matrix with a different norm. Using polynomial methods derived from the kernel conditions, we establish that any element is necessarily nilpotent with same order of the truncated shift. These results reveal that the Harnack equivalence class exhibits a significantly richer algebraic structure in dimension five than previously observed in lower-dimensional cases.