A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.