Fixed points and normal automorphisms of the unit ball of bounded operators on $\mathbb{C}^n$

TL;DR

This paper characterizes the normal isometries of the open unit ball of bounded linear operators on complex Banach spaces, focusing on their fixed points and the structure of the isometry group.

Contribution

It provides a new characterization of normal automorphisms of the operator unit ball using fixed point properties, advancing understanding of their geometric structure.

Findings

Normal isometries are characterized by their fixed points.

The structure of the isometry group is described in terms of fixed point sets.

The Carathéodory metric plays a key role in the analysis.

Abstract

We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carath\'eodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.