A condition for the convexity of the norm-numerical range of a matrix

AbstractLet v be a norm on Cn, and let a be a matrix which has a v-Hermitian decomposition. (1) The v-numerical range of a is convex. (This generalizes the Hausdorff-Toeplitz theorem.) In fact, the v-numerical range is equal to the field of values of a matrix similar to a. (2) If the Hermitian and v-Hermitian decompositions of a coincide, then the v-numerical range of a and the field of values of a are the same. This follows from detailed information about the boundary of the range