Indefinite higher-rank numerical ranges

Abstract

We investigate indefinite higher-rank numerical ranges of a wide class of J-Hermitian matrices, J = I _r  ⊕ −I _n−r , 0 < r < n (A ∈ C ^n×n is said to be J-Hermitian if A = JA*J). Particular attention is paid to aspects of the theory that parallel the case of Hermitian matrices.

Keywords:

• indefinite inner product
• indefinite higher-rank numerical range
• classical numerical range
• J-Hermitian matrix
• J-normal matrix

AMS Subject Classifications::

• 47A12
• 15A60

Acknowledgements

The authors wish to thank Professor H. Nakazato for helpful discussions. Thanks are also due to the referees for their valuable comments and for careful reading of this article. The present research was partially supported by Project PTDC/MAT/69613/2006.