Krein Space Numerical Range of Block Matrices -- a Unified Approach to the Hyperbolic Case

TL;DR

This paper studies the Krein space numerical range of 2x2 block matrices with scalar identity blocks, focusing on hyperbolic boundary curves to unify and extend existing results about their shape.

Contribution

It introduces a unified approach to analyze the hyperbolic shape of the numerical range for specific block matrices, including new findings.

Findings

Characterization of the numerical range boundary as hyperbolas

Unified framework for existing and new results

Extension to broader classes of block matrices

Abstract

In this paper, we investigate the Krein space numerical range of $2$-by-$2$ block matrices, with diagonal blocks as scalar multiples of the identity. For these matrices, we specifically investigate the cases when the respective boundary generating curves consist of hyperbolas. This provides a unified approach to derive established and new results concerning the numerical range hyperbolic shape.