Group of isometries of the Hilbert ball equipped with the Caratheodory metric

TL;DR

This paper investigates the geometric structure of an infinite-dimensional hyperbolic space, focusing on the isometry group of the Hilbert ball with the Carathéodory metric, including subclasses, unitary conditions, and cardinalities.

Contribution

It characterizes the isometry group of the Hilbert ball with the Carathéodory metric and identifies special subclasses and conditions for unitary equivalence.

Findings

Identified subclasses of the isometry group

Derived conditions for unitary equivalence

Computed cardinalities of certain subclasses

Abstract

In this article, we study the geometry of an infinite dimensional Hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carath$\acute{e}$odory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities.