Hyperbolic geometry on noncommutative balls

TL;DR

This paper explores the hyperbolic geometry of noncommutative balls generated by various joint operator radii, establishing key inequalities and lemmas for free holomorphic functions in this setting.

Contribution

It introduces a geometric framework for noncommutative balls using joint operator radii and proves fundamental inequalities and lemmas for free holomorphic functions.

Findings

Mapping theorems for noncommutative balls

Von Neumann inequalities in hyperbolic geometry

Schwarz type lemmas for free holomorphic functions

Abstract

In this paper, we study the hyperbolic geometry of noncommutative balls generated by the joint operator radius $\omega_\rho$, $\rho\in (0,\infty]$, for $n$-tuples of bounded linear operators on a Hilbert space. In particular, $\omega_1$ is the operator norm, $\omega_2$ is the joint numerical radius, and $\omega_\infty$ is the joint spectral radius. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric $\delta_\rho$, the Carath\' eodory metric $d_K$, and the joint operator radius $\omega_\rho$.