Invariant subspaces for non-normable Fr\'echet spaces

TL;DR

This paper investigates the invariant subspace property in non-normable Fréchet spaces, identifying conditions under which such spaces have or lack this property, including examples involving the space of entire functions.

Contribution

It constructs examples of non-normable Fréchet spaces with the hereditary invariant subspace property and provides conditions for operators without invariant subsets, including on $H(\mathbb{C})$.

Findings

Certain non-normable Fréchet spaces satisfy the hereditary invariant subspace property.

Many non-normable Fréchet spaces do not satisfy this property.

There exists a continuous operator on $H(\mathbb{C})$ without non-trivial invariant subsets.

Abstract

A Fr\'echet space $X$ satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace $M$ in $X$, each continuous operator on $M$ possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fr\'echet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fr\'echet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce among other examples that there exists a continuous operator without non-trivial invariant subset on the space of entire functions $H(\mathbb{C})$.