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

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 $\epsilon >0$ such that, for every $a\in (0,\epsilon )$, one can find a rank-four partial isometry $V_a\in {\text{M}}_6(\mathbb{R})$ satisfying $$W(V_a)=\{\zeta \in \text{C}:\mathrm{\mid }\zeta -a\mathrm{\mid }\le 3\mathrm{/}4\}\mathrm{.}$$