Absolutely compatibility in $\mathbb{M}_2$ and ellipticity

TL;DR

This paper characterizes all absolutely compatible pairs of elements in 2x2 matrices over complex numbers, revealing that non-commutative pairs display an elliptic pattern, thus advancing understanding of compatibility in operator algebras.

Contribution

It provides a complete classification of absolutely compatible pairs in 2 matrices, highlighting the elliptic behavior of non-commutative pairs.

Findings

Complete list of absolutely compatible pairs in 2

Non-commutative pairs exhibit elliptic behavior

Advances understanding of compatibility in operator algebras

Abstract

Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note, we describe a complete list of absolutely compatible pairs in $\mathbb{M}_2$. We note that non-commutative pairs exhibit an elliptic behaviour.