On the numerical range map

Let A is an element of L(C-n) and A(1), A(2) be the unique Hermitian operat ors such that A = A(1) + iA(2). The paper is concerned with the differentia l structure of the numerical range map n(A) : x --> ([A(1)x,x], [A(2)x, x]) and its connection with certain natural subsets of the numerical range W(A ) of A. We completely characterize the various sets of critical and regular points of the map n(A) as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippen hahn appear naturally in this context. They basically coincide with the ima ge of the critical points of n(A).