Orbital and strongly orbital spaces

We say that a (countably dimensional) topological vector space X is orbital if there is T∈L(X) and a vector x∈X such that X is the linear span of the orbit {Tnx: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∈L(X) such that every non-zero x∈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=ℓ2×ω, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators T1,T2,…,TN with N⩾2 which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators T1, T2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.