The star-shapedness of a generalized numerical range

Abstract

Let ${\mathcal{H}}_n$ be the set of all $n\times n$ Hermitian matrices and ${\mathcal{H}}_n^m$ be the set of all $m$-tuples of $n\times n$ Hermitian matrices. For $A=(A_1,...,A_m)\in {\mathcal{H}}_n^m$ and for any linear map $L:{\mathcal{H}}_n^m\to {\mathbb{R}}^{\ell }$, we define the $L$-numerical range of $A$ by \[ W_L(A):=\{L(U^*A_1U,...,U^*A_mU): U\in \mathbb{C}^{n\times n}, U^*U=I_n\}. \] In this paper, we prove that if $\ell \le 3$, $n\ge \ell$ and $A_1,...,A_m$ are simultaneously unitarily diagonalizable, then $W_L(A)$ is star-shaped with star center at $L\left(\frac{\mathrm{tr}{A}_1}{n}{I}_n,...,\frac{\mathrm{tr}{A}_m}{n}{I}_n\right)$.