The Strong Disjoint Blow-UpCollapse Property

Abstract EN

Let X be a topological vector space, and let ℬ(X) be the algebra of continuous linear operators on X.

The operators T1,…,TN∈ℬ(X) are disjoint hypercyclic if there is x∈X such that the orbit {(T1n(x),…,TNn(x)):n∈ℕ} is dense in X×…×X.

Bès and Peris have shown that if T1,…,TN satisfy the Disjoint Blow-up/Collapse property, then they are disjoint hypercyclic.

In a recent paper Bès, Martin, and Sanders, among other things, have characterized disjoint hypercyclic N-tuples of weighted shifts in terms of this property.

We introduce the Strong Disjoint Blow-up/Collapse property and prove that if T1,…,TN satisfy this new property, then they have a dense linear manifold of disjoint hypercyclic vectors.

This allows us to give a partial affirmative answer to one of their questions.