Orbital and strongly orbital 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 Fréchet space $X$. For instance, in $X=\ell_2\times ω$, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.