The boundary of the range of a constrained sesquilinear form

Abstract

Let A be a n×n complex matrixq be a complex number with |q|< 1. We denote by W(A:q) the compact subset {y^*Ax:xεCⁿ, yεCⁿ x^*x=1y^*y=1,y^*x=q of C and call it the q-numerical range of A. In this paper we show that if A is a normal matrix and |q| < 1, then the boundary of the compact convex set W(A:q) is the union of a finite number of algebraic arcs and every such arc lies on the boundary of an elliptical disc with eccentricity |q| or 0.