# Hypercyclic bilinear operators on Banach spaces

## TL;DR

This paper demonstrates the existence of hypercyclic bilinear operators on Banach spaces, providing new examples and answering open questions about their dynamics and properties.

## Contribution

It introduces the first known examples of m-linear hypercyclic operators on Banach spaces and addresses questions about symmetric bihypercyclic bilinear operators.

## Key findings

- Existence of m-linear hypercyclic operators on any infinite dimensional separable Banach space
- Construction of a specific m-linear hypercyclic operator
- Existence of symmetric bihypercyclic bilinear operators

## Abstract

We study the dynamics induced by an $m$-linear operator. We answer a question of B\`es and Conejero showing an example of an $m$-linear hypercyclic operator acting on a Banach space. Moreover we prove the existence of $m$-linear hypercyclic operators on arbitrary infinite dimensional separable Banach spaces. We also prove an existence result about symmetric bihypercyclic bilinear operators, answering a question by Grosse-Erdman and Kim.