Invariant subspaces for Fr\'echet spaces without continuous norm

TL;DR

This paper characterizes when certain Fréchet spaces without continuous norms have the invariant subspace property, linking it to the finite codimension of kernels of increasing seminorms.

Contribution

It provides a necessary and sufficient condition for the invariant subspace property in Fréchet spaces with Schauder bases and no continuous norm.

Findings

Invariant subspace property holds iff kernels of seminorms have finite codimension

Characterization applies to Fréchet spaces with Schauder basis and no continuous norm

Condition involves the structure of kernels of increasing seminorms

Abstract

Let $(X,(p_j))$ be a Fr\'echet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and only if there exists $j_0\ge 1$ such that $\ker p_{j+1}$ is of finite codimension in $\ker p_{j}$ for every $j\ge j_0$.