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

TL;DR

This paper establishes a canonical form for pairs of a nondegenerate Hermitian form and a self-adjoint antilinear operator, generalizing previous diagonalization results and aiding in the study of CR structures in differential geometry.

Contribution

It introduces a canonical form for such pairs, extending prior results limited to positive definite cases and providing a unified framework.

Findings

Unified canonical form for pairs of Hermitian and self-adjoint antilinear operators

Generalization of diagonalization criteria beyond positive definite cases

Application to local differential geometry of CR structures

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 Benedetti and Cragnolini (1984) on simultaneous diagonalization of these pairs in the case where the Hermitian form is positive definite and of Hong and Horn (1986), where a criterion for simultaneous diagonalization is given.