The J-numerical range of a J-Hermitian matrix and related inequalities

Recently, indefinite versions of classical inequalities of Schur, Ky Fan and Rayleigh–Ritz on Hermitian matrices have been obtained for J-Hermitian matrices that are J-unitarily diagonalizable, J=Ir⊕(-Is),r,s>0. The inequalities were obtained in the context of the theory of numerical ranges of operators on indefinite inner product spaces. In this paper, the subject is revisited, relaxing the constraint of the matrices being J-unitarily diagonalizable.