Geometry of the numerical range of Krein space operators

Abstract

The characteristic polynomial of the pencil generated by two J-Hermitian matrices is studied in connection with the numerical range. Geometric properties of the numerical range of linear operators on an inde nite inner product space are investigated. The point equation of the associated curve of the numerical range is derived, following Fiedler's approach for de nite inner product spaces. The classi cation of the associated curve in the 3 £ 3 case is presented, using Newton's classi cation of cubic curves. As a consequence, the respective numerical ranges are characterized. Illustrative examples of all the di erent possibilities are given.