The Gau-Wang-Wu conjecture on partial isometries holds in the 5-by-5 case

TL;DR

This paper proves that the Gau-Wang-Wu conjecture, which states that an n-by-n partial isometry cannot have a circular numerical range with a non-zero center, holds true for the case n=5.

Contribution

The paper extends the proof of the Gau-Wang-Wu conjecture to the 5-by-5 case, filling the gap between previous cases and the general conjecture.

Findings

Confirmed the conjecture for n=5

Partial isometries in 5x5 matrices do not have circular numerical ranges with non-zero centers

Supports the conjecture's validity for all n ≥ 1

Abstract

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.