Numerical ranges of Foguel operators revisited

The Foguel operator is defined as $F_T=\begin{bmatrix}S^* & T \ 0 & S\end{bmatrix}$,where$S$istherightshiftonaHilbertspace$\mathcal H$and$T$canbeanarbitraryboundedlinearoperatoractingon$\mathcal H$. Obviously, the numerical range $W(F_0)$ of $F_T$ with $T=0$ is the open unit disk, and it was suggested by Gau, Wang and Wu in their LAA’2021 paper that $W(F_{aI})$ for non-zero $a\in \mathbb{C}$ might be an elliptical disk. In this paper, we described $W(F_{aI})$ explicitly and, as it happens, it is not.