On the numerical range map. (English) Zbl 0932.15016

Authors’ abstract: Let \(A\in {\mathcal L}({\mathbb C}^n)\) and \(A_1\), \(A_2\) be the unique Hermitian operators such that \(A=A_1+iA_2\). The paper is connected with the differential structure of the numerical range map \(n_A: x\mapsto (\langle A_1x,x\rangle, \langle A_2x,x\rangle)\) 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. Kippenhahn [Math. Nachr. 6, 193-228 (1951; Zbl 0044.16201)] appear naturally in this context. They basically coincide with the image of the critical points of \(n_A\).


15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A22 Matrix pencils


Zbl 0044.16201