Operators approximable by hypercyclic operators

TL;DR

This paper proves that operators of the form I + S, with S having a dense generalized kernel, can be approximated by hypercyclic and mixing operators on separable infinite-dimensional Banach spaces.

Contribution

It establishes that such operators are in the norm closure of hypercyclic and mixing operators, advancing understanding of operator approximations in Banach spaces.

Findings

Operators of the form I + S with dense generalized kernel are in the closure of hypercyclic operators.

These operators are also in the closure of mixing operators.

The result applies to operators on separable infinite-dimensional Banach spaces.

Abstract

We show that operators on a separable infinite dimensional Banach space $X$ of the form $I +S$, where $S$ is an operator with dense generalised kernel, must lie in the norm closure of the hypercyclic operators on $X$, in fact in the closure of the mixing operators.