On the boundary of the<i>C</i>-numerical range of a matrix

Abstract

Let C and A be n × n complex matrices. The C-numerical range of A is the set Wc (A) = {tr(CUAU∗) unitary}in. Given c = (c1 …cn )∊ Cn, the set Wc (A) is denoted by Wc (A) and said to be the c-numerical range in the case that C is the diagonal matrix with diagonal entries c = (c1 ,…,cn ). In this paper we study the boundary ∂Wc (A) of Wc (A). Above all, we show the following:A non-differentiable point of ∂Wc (A) is a pivot of a sector which is formed by ∂Wc (A), in the case of c =(c1 … cn ). All differentiable points of ∂Wc (A) are classified via their degrees of smoothness. For example, there exists a case in which a C1 -smooth point of ∂W(1,0,…0) (A) is not analytically smooth. However, the number of non-analytically smooth points of ∂Wc (A) is at most finite.