On the hyperbolicity of the Krein space numerical range

Abstract

In this paper a necessary and sufficient condition for hyperbolicity of the indefinite numerical range is established. As a consequence, an indefinite version of Brown–Spitkovsky theorem stating the ellipticity of the numerical range of certain tridiagonal matrices is revisited. This result leads to necessary and sufficient conditions for hyperbolicity of indefinite numerical ranges of new classes of tridiagonal matrices.

Keywords:

• Indefinite numerical range
• classical numerical range
• J-Hermitian matrix
• hyperbolical range theorem

AMS Classifications:

• 47A12
• 15A60

Acknowledgments

The authors would like to thank the referee for careful reading of the manuscript and comments which helped improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of the first author was partially supported by the CMUC (Centre for Mathematics of the University of Coimbra) [grant number UIDB/00324/2020], funded by the Portuguese Government through FCT/MCTES. The second author was supported by the Portuguese funds through the CIDMA (Center for Research and Development in Mathematics and Applications) and the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia, I.P.) [project number UIDB/04106/2020]. The third author was financed by the CMAT-UTAD and Portuguese funds through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia, I.P.) [reference number UIDB/00013/2020].