Decomposable numerical ranges of normal matrices

Let $M_{n,k}$ ($M_n$) be the set of $n\times k$ ($n\times n$) complex matrices, and $\mathrm{per}\left(X\right)$ be the permanent of a square matrix X. We study the three types of generalized numerical ranges associated with generalized matrix functions${\Pi }_k\left(A\right)=\{\prod _{j=1}^k{\left(V^{⁎}AV\right)}_{ii}:V\in M_{n,k},V^{⁎}V=I_k\},$$D_k\left(A\right)=\{\det (V^{⁎}AV):V\in M_{n,k},V^{⁎}V=I_k\},$ and$P_k\left(A\right)=\left\{\mathrm{per}\right(V^{⁎}AV):V\in M_{n,k},V^{⁎}V=I_k\}.$ We give complete descriptions of the set ${\Pi }_2\left(A\right)$, $D_2\left(A\right)$ and $P_2\left(A\right)$ for essentially hermitian matrices $A\in M_n$. In particular, all three sets are star-shaped. For $3\times 3$ normal matrices A, it is known that $D_2\left(A\right)$ is convex. We show that ${\Pi }_2\left(A\right)$ and $P_2\left(A\right)$ are star-shaped. This affirms a conjecture of Nakazato et al.