The generalized numerical range of a set of matrices

TL;DR

This paper explores the properties of the generalized numerical range of a set of matrices, revealing its algebraic and geometric characteristics, including conditions for convexity and star-shapedness, and extending to joint numerical ranges.

Contribution

It introduces new algebraic and topological properties of the generalized numerical range and establishes conditions for convexity and star-shapedness, extending to joint numerical ranges.

Findings

W_C({ extstyle ext{F}}) has specific algebraic and topological properties.

Convexity of W_C(A) and W_C(B) implies star-shapedness of W_C({ extstyle ext{F}}).

Results extend to joint C-numerical ranges of matrix tuples.

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.