A Counterexample to the Gau--Wang--Wu Conjecture on Partial Isometries

Abstract

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{\zeta\in\C:|\zeta-a|\leq 3/4\}. $$