The star-shapedness of a generalized numerical range

TL;DR

This paper proves that the generalized numerical range of certain Hermitian matrix tuples is star-shaped when the linear map has dimension up to three, under conditions of simultaneous diagonalizability.

Contribution

It establishes star-shapedness of the generalized numerical range for Hermitian matrices under specific dimensional and diagonalizability conditions.

Findings

The $L$-numerical range is star-shaped for $ ext{dim}(L)\leq 3$.

Star center is the image of the scaled trace of matrices.

Results extend understanding of geometric properties of matrix ranges.

Abstract

Let $\mathcal{H}_n$ be the set of all $n\times n$ Hermitian matrices and $\mathcal{H}^m_n$ be the set of all $m$-tuples of $n\times n$ Hermitian matrices. For $A=(A_1,...,A_m)\in \mathcal{H}^m_n$ and for any linear map $L:\mathcal{H}^m_n\to\mathbb{R}^\ell$, 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 $\ell\leq 3$, $n\geq \ell$ and $A_1,...,A_m$ are simultaneously unitarily diagonalizable, then $W_L(A)$ is star-shaped with star center at $L\left(\frac{\mathrm{tr} A_1}{n}I_n,...,\frac{\mathrm{tr} A_m}{n}I_n\right)$.