On the boundary of weighted numerical ranges

Abstract

In this article, we introduce the weighted numerical range which is a unified approach to study the -numerical range and the rank numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of different sizes) overlap at sufficiently many points, then the two matrices share common generalized eigenvalues.

Keywords:

• numerical ranges
• c-numerical ranges
• rank k numerical ranges
• Bezout’s Theorem

AMS Subject Classifications:

• 15A60
• 15A42
• 15A18

Acknowledgements

I would like to thank C.K. Li and Karol Życzkowski for their valuable suggestions. I would also like to thank the referee for pointing out a mistake in the first version of Lemma 4.2 and other errors.

Notes

No potential conflict of interest was reported by the author.