Hypercyclic subspaces on Fr\'echet spaces without continuous norm

TL;DR

This paper explores the existence and characteristics of hypercyclic subspaces in Fréchet spaces lacking a continuous norm, extending known results to a broader class of spaces.

Contribution

It introduces a classification of hypercyclic subspaces in such spaces and provides criteria for their existence, generalizing previous findings.

Findings

Every infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.

Established criteria for two types of hypercyclic subspaces in spaces without continuous norm.

Generalized results from spaces with continuous norms to a wider class of Fréchet spaces.

Abstract

Known results about hypercyclic subspaces concern either Fr\'echet spaces with a continuous norm or the space \omega. We fill the gap between these spaces by investigating Fr\'echet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that M\cap \ker p=\{0\} and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fr\'echet spaces with a continuous norm and about hypercyclic subspaces on \omega. In particular, we show that each infinite-dimensional separable Fr\'echet space supports a mixing operator with a hypercyclic subspace.