Journal of the Australian Mathematical Society - Series A Vol. 65 Part 2 (1998) On the numerical range map M. Joswig Fachbereich Mathematik Technische Universität Berlin Strasse des 17. Juni 136 D-10623 Berlin Germany and B. Straub School of Mathematics The University of New South Wales Sydney, NSW 2052 Australia Abstract: Let $A\in\mathcal{L}(\mathbb{C}^n)$ and A1, A2 be the unique Hermitian operators such that A=A1+iA2. The paper is concerned with the differential structure of the numerical range map $n_A:x\longmapsto \left(\langle A_1x,x\rangle,\langle A_2x,x\rangle\right)$ 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 nA as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippenhahn appear naturally in this context. They basically coincide with the image of the critical points of nA. PDF file size: 134K © Copyright 1998, Australian Mathematical Society