An inverse indefinite numerical range problem

Consider CnCn with a Krein space structure with respect to the indefinite inner product [x,y]=x⁎Jy[x,y]=x⁎Jy, x,y∈Cnx,y∈Cn, where J   is an indefinite self-adjoint involution. The Krein space numerical range WJ(T)WJ(T) of a complex matrix T   is the set of all the values attained by the quadratic form [Tu,u][Tu,u], where u∈Cnu∈Cn satisfies [u,u]=±1[u,u]=±1. The main aim of this paper is the investigation of the following inverse problem: given a complex matrix T and a point z   in WJ(T)WJ(T), determine a unit vector that generates z. The number of linearly independent generating vectors of z is determined. An algorithm for solving the inverse problem is developed, implemented and tested.