The generalized numerical range of a set of matrices

Abstract

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