A definition of numerical range of rectangular matrices

Abstract

Bonsall and Duncan (1973) observed that the numerical range of a bounded linear operator can be written as an infinite intersection of closed circular discs. Motivated by this interesting property (which does not seem to be very popular with people working on numerical ranges), we propose a definition of numerical range of rectangular complex matrices. The new range is always compact and convex, and satisfies basic properties of the standard numerical range. Our analysis is based on the properties of norms and the Birkhoff–James orthogonality.

Keywords:

• numerical range
• norm
• Birkhoff–James orthogonality

AMS subject classifications: :

• 15A60
• 47A12

Acknowledgements

The authors acknowledge with thanks an anonymous referee for his/her useful suggestions. Research supported by a grant of the EPEAEK project PYTHAGORAS II. The project is co-funded by the European Social Fund (75%) and Greek National Resources (25%).