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arXiv · 1805.00602

The generalized numerical range of a set of matrices

Pan-Shun Lau·Chi-Kwong Li·Yiu-Tung Poon·Nung-Sing Sze
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Abstract

For a given set of $n\times n$ matrices $\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\mathcal F$, denoted by $W_C({\mathcal F})$. We obtain basic algebraic and topological properties of $W_C({\mathcal F})$, and show that there are connections between the geometric properties of $W_C({\mathcal F})$ and the algebraic properties of $C$ and the matrices in ${\mathcal F}$. Furthermore, we consider the starshapedness and convexity of the set $W_C({\mathcal F})$. In particular, we show that if ${\mathcal F}$ is the convex hull of two matrices such that $W_C(A)$ and $W_C(B)$ are convex, then the set $W_C({\mathcal F})$ is star-shaped. We also investigate the extensions of the results to the joint $C$-numerical range of an $m$-tuple of matrices.

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Pan-Shun Lau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze. 2018-05-02. The generalized numerical range of a set of matrices. https://arxiv.org/abs/1805.00602

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