A canonical form for pairs consisting of a Hermitian form and a self-adjoint antilinear operator

Abstract

Motivated by a problem in local differential geometry of Cauchy–Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or, equivalently, consisting of a nondegenerate Hermitian form and a symmetric bilinear form. This generalizes the only previously known results on simultaneous normalization of such pairs, namely, the results of [2] on simultaneous diagonalization of these pairs in the case where the Hermitian form is positive definite and of [11], where a criterion for simultaneous diagonalization is given.