# Linear maps preserving inner functions

## TL;DR

This paper demonstrates that in many holomorphic function spaces on the unit disk, any continuous endomorphism that preserves inner functions must be a weighted composition operator, revealing a structural characterization.

## Contribution

It establishes a general result linking inner function preservation to weighted composition operators in various holomorphic spaces.

## Key findings

- Inner function-preserving endomorphisms are weighted composition operators
- The result applies to many holomorphic function spaces on the unit disk
- Provides a structural understanding of such endomorphisms

## Abstract

We show that, for many holomorphic function spaces on the unit disk, a continuous endomorphism that sends inner functions to inner functions is necessarily a weighted composition operator.