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

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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.

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