Orbital and strongly orbital spaces

TL;DR

This paper characterizes orbital and strongly orbital topological vector spaces, showing their properties, constructing examples under the Continuum Hypothesis, and analyzing the structure of dense countably dimensional subspaces.

Contribution

It provides a complete characterization of orbital and strongly orbital metrizable locally convex spaces and constructs examples assuming the Continuum Hypothesis.

Findings

Characterization of orbital and strongly orbital spaces

Existence of spaces without the invariant subset property

Classification of dense countably dimensional subspaces in certain spaces

Abstract

We say that a (countably dimensional) topological vector space $X$ is orbital if there is $T\in L(X)$ and a vector $x\in X$ such that $X$ is the linear span of the orbit ${T^nx:n=0,1,...}$. We say that $X$ is strongly orbital if, additionally, $x$ can be chosen to be a hypercyclic vector for $T$. Of course, $X$ can be orbital only if the algebraic dimension of $X$ is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space $X$ does not have the invariant subset property. That is, there is $T\in L(X)$ such that every non-zero $x\in X$ is a hypercyclic vector for $T$. Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space. As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fr\'echet space $X$. For instance, in $X=\ell_2\times \omega$, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.