The numerical range of matrix products

We discuss what can be said about the numerical range of the matrix product A(1 )A(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)) subset of K-j, it follows that W(A(1) A(2)) subset of K. We do this by defining specific deviation quantities for both the unrestricted case and the normal case.