Surjective isometries on Banach sequence spaces: a survey

TL;DR

This survey reviews the characterization of surjective isometries on Banach sequence spaces, covering classical, combinatorial, and Tsirelson-type spaces, and discusses open problems in the structure of their isometry groups.

Contribution

It compiles and proves key results on surjective isometries across various Banach sequence spaces, including recent developments and open questions.

Findings

Complete characterizations for classical spaces

Recent results on combinatorial and Tsirelson-type spaces

Open problems on isometry group structures

Abstract

In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries and characterizations of the group of surjective isometries for various Banach spaces.