# Surjective isometries on a Banach space of analytic functions with bounded derivatives

## TL;DR

This paper characterizes surjective isometries on a space of analytic functions with bounded derivatives, extending previous work on linear isometries in Hardy spaces to a broader class of isometries.

## Contribution

It provides a complete description of surjective isometries on the space of analytic functions with bounded derivatives, generalizing known results for linear isometries.

## Key findings

- Characterization of surjective isometries on the space of functions with bounded derivatives.
- Extension of previous linear isometry results to non-linear isometries.
- New insights into the structure of isometries in analytic function spaces.

## Abstract

Let $H(\mathbb{D})$ be the linear space of all analytic functions on the open unit disc $\mathbb{D}$ and $H^p(\mathbb{D})$ the Hardy space on $\mathbb{D}$. The characterization of complex linear isometries on $\mathcal{S}^p=\{f\in H(\mathbb{D}):f'\in H^p(\mathbb{D})\}$ was given for $1\leq p<\infty$ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on $\mathcal{S}^\infty$.