The Strong Disjoint Blow-Up/Collapse Property

Let be a topological vector space, and let be the algebra of continuous linear operators on . The operators are disjoint hypercyclic if there is such that the orbit is dense in . Bès and Peris have shown that if 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 -tuples of weighted shifts in terms of this property. We introduce the Strong Disjoint Blow-up/Collapse property and prove that if 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. 1. Introduction and Background Let be a topological vector space, over either the real or complex numbers, whose topology has a countable basis and is complete. Let be the algebra of continuous linear operators on . The operator is hypercyclic if there is such that Orb is dense in . This concept is closely related to the concept of transitivity from topological dynamics. Definition A. The operator is topologically transitive if for each pair , of nonempty open subsets of there is such that . In fact, both notions are equivalent in our setting. This is the content of Birkhoff's Transitivity Theorem; see for instance 1.7 of the instructive notes by Shapiro [1]. The first version of the Hypercyclicity Criterion, whose importance is that if an operator satisfies it then it is hypercyclic, was given by Kitai in [2] and by Gethner and Shapiro in [3]. Several mathematicians had given different versions of it. One of them is the following. Definition B. The operator satisfies the Blow-up/Collapse property if whenever nonempty open sets , , are given with neighbourhood of 0, then there exits such that This suggestive name was coined by Grosse-Erdmann who used it in several talks that he gave years ago. The concept itself was introduced by Godefroy and Shapiro, who showed that it is implied by the Hypercyclicity Criterion [4]. Bernal-González and Grosse-Erdmann [5] and León-Saavedra [6] showed, independently, the other implication. Thus satisfies the Blow-up/Collapse property if and only if satisfies the Hypercyclicity Criterion. For a long time all known hypercyclic operators were known to satisfy some version of the Hypercyclicity Criterion. A milestone paper by de la Rosa and Read [7] showed that this is not always the case. The excellent books by Bayart and Matheron [8] and Grosse-Erdmann and Peris [9] provide a solid foundation and give an overview of much of the