Computation of Krein space numerical ranges of 2x2 matrices

Abstract

In this paper, we investigate some properties of the Krein space numerical range and compute the Krein space $J$-numerical ranges of some $2 \times 2$ complex matrices with respect to several fundamental symmtries.
We also compare the direct computation of Krein space numerical ranges with the computation via the joint numerical ranges.
Finally, we discuss the Krein space numerical ranges of upper triangular matrices.