On the circular numerical range of 5-by-5 partial isometries

TL;DR

This paper investigates the conditions under which 5-by-5 partial isometries have a circular numerical range, proving that if the numerical range is a circle, its center must be at the origin, partially confirming a conjecture.

Contribution

It provides a partial proof of a conjecture regarding the center of the numerical range of 5-by-5 partial isometries when the range is circular.

Findings

If the numerical range of a 5-by-5 partial isometry is a circle, its center is at the origin.

The paper offers a partial affirmative answer to a conjecture in the five-dimensional case.

Uses Kippenhahn curves to analyze the numerical range properties.

Abstract

We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center is must be the origin. This gives a partial affirmative answer of the Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14--35.], for the five dimensional case.