The invariant subspace problem for the space of smooth functions on the real line

TL;DR

This paper constructs a continuous linear operator on the space of smooth functions on the real line that lacks non-trivial invariant subspaces, marking a novel example in a Fréchet space without a continuous norm.

Contribution

It provides the first example of such an operator on a Fréchet space without a continuous norm, extending Read's ideas from Banach spaces to this setting.

Findings

Constructed a continuous operator on smooth functions with no non-trivial invariant subspaces.

First example of such an operator on a Fréchet space without a continuous norm.

Extends previous work by C. Read to a new class of function spaces.

Abstract

We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fr\'echet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space $\ell_1$.