The elliptical range theorem for the conformal range

TL;DR

This paper extends the elliptical range theorem to the conformal range, a hyperbolic analogue of the numerical range, providing new insights into the geometric properties of $2\times2$ matrices in hyperbolic space.

Contribution

It establishes an analogue of the elliptical range theorem specifically for the conformal range of $2\times2$ matrices, linking matrix theory with hyperbolic geometry.

Findings

The conformal range can be viewed as a hyperbolic field of values.

An elliptical range theorem analogue is proven for the conformal range.

The results connect matrix analysis with hyperbolic geometric structures.

Abstract

The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of $2\times2$ complex matrices for the conformal range.