The numerical range of matrix products

Abstract

We discuss what can be said about the numerical range of the matrix product $A_1A_2$ when the numerical ranges of $A_1$ and $A_2$ are known. If two compact convex subsets $K_1, K_2$ of the complex plane are given, we discuss the issue of finding a compact convex subset $K$ such that whenever $A_j$ ($j=1,2$) are either unrestricted matrices or normal matrices of the same shape with $W(A_j) \subseteq K_j$, it follows that $W(A_1A_2) \subseteq K$. We do this by defining specific deviation quantities for both the unrestricted case and the normal case.