DIFFERENTIAL PROPERTIES OF THE NUMERICAL RANGE MAP OF PAIRS OF MATRICES

Let A = (A(1), A(2)) be a pair of Hermitian operators in C-n and A = A (1) + iA(2). We investigate certain differential properties of the num erical range map n(A) : x --> ([A(1)x, x], [A(2)x, x]) with the aim of better understanding the nature of the numerical range W(A) of A. For example, the joint eigenvalues of A correspond to the stationary poin ts of n(A) (i.e. points where the derivative n'(A) vanishes). Moreover , points x where rank n'(A)(x) = 2 get mapped by n(A) into the interio r W(A)degrees of W(A). For n = 2, it turns out that if A(1) and A(2) h ave no common invariant subspace, then the image under n(A) of the set Sigma(1)( A) consisting of those points x with rank n'(A)(x) = 1 is p recisely the boundary partial derivative W(A) of W(A), and the image u nder n(A) of the rank 2 points for n'(A) is precisely W(A)degrees; the re are no rank 0 points for n'(A). As a consequence (for n = 2) we hav e that A(1)A(2) = A(2)A(1) iff Sigma(1)(A) not equal n(A)(-1)(partial derivative W(A)). (C) 1997 Elsevier Science Inc.