# Linear operators with infinite entropy

## TL;DR

This paper investigates the chaotic dynamics of specific linear operators on infinite-dimensional Banach spaces, establishing conditions for infinite topological entropy and answering existing questions about their entropy properties.

## Contribution

It provides new characterizations of when these operators have infinite entropy and demonstrates that certain translation operators cannot have finite non-zero entropy.

## Key findings

- Infinite topological entropy is characterized by specific operator properties.
- Translation operators on weighted Lebesgue spaces always have infinite entropy.
- Finite non-zero entropy is impossible for these translation operators.

## Abstract

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite topological entropy is equivalent to non-zero topological entropy for translation operators on weighted Lebesgue function spaces. In particular, finite non-zero entropy is impossible for this class of operators, which answers a question raised by Yin and Wei.