The Strong Disjoint Blow-Up / Collapse Property

N (x)) : n ∈ N} is dense in X × . . . × X. Bès and Peris have shown that if T 1 , . . . , T N 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 T 1 , . . . , T N 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.


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 B() be the algebra of continuous linear operators on .
The operator  ∈ B() is hypercyclic if there is  ∈  such that Orb(, ) : = {   :  ∈ N} is dense in .This concept is closely related to the concept of transitivity from topological dynamics.
Definition A. The operator  ∈ B() is topologically transitive if for each pair ,  of nonempty open subsets of  there is  such that  ∩  − () ̸ = 0.
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  ∈ B() satisfies the Blowup/Collapse property if whenever nonempty open sets ,  0 ,  1 are given with  neighbourhood of 0, then there exits  such that  − () ∩  0 ̸ = 0,  − ( 1 ) ∩  ̸ = 0.
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 work done in hypercyclicity.The Blow-up/Collapse property is mentioned in page 85 of [9].The following concept was introduced independently by Bernal-González [10] and Bès and Peris [11].It is worth noting that while the author of [10] was inspired by some recent work by Costakis and Vlachou in universal Taylor series, the authors of [11] were inspired by much older work of Furstenberg for dynamical systems in which he studied the notion of disjointness, "an extreme form of nonisomorphism" according to Parry.
The following three definitions were given in [11] in a slightly more general way.The first one is their Definition 2.1.
The second one is included in their Proposition 2.4.
The third one is their Definition 2.5.(If  were allowed to be 1, then this would be one of the many versions of the classical Hypercyclicity Criterion.)Definition F. Let  ≥ 2. Let (  ) be a strictly increasing sequence of positive integers.The operators  1 , . . .,   ∈ B() satisfy the Disjoint Hypercyclicity Criterion with respect to (  ) provided there exist dense subsets  0 ,  1 , . . .,   of  and mappings  , :   →  with 1 ≤  ≤  and  ∈ N satisfying In the last display    means the identity in   .(It has to be   and not   because   is the domain of  , .)We now give some known relations between all these concepts.Proposition 2.3 of [11] says that  1 , . . .,   ∈ B() are disjoint topologically transitive if and only if the set of disjoint hypercyclic vectors for  1 , . . .,   is a dense   of .
Proposition 2.4 of [11] says that if  1 , . . .,   ∈ B() satisfy the Disjoint Blow-up/Collapse property, then they are disjoint topologically transitive.This is the disjoint version of Theorem 1.2 of [4], but the authors of [4] assume that  is a Banach space.Theorem 2.7 of [11] [18].The relative simplicity with which the authors of [18] show disjoint hypercyclic operators which do not satisfy the Disjoint Hypercyclicity Criterion should be contrasted with the sophistication of the arguments in [7].
It is our goal in this paper to study a "strong" version of the Blow-up/Collapse property.We show that for the class of -tuples of weighted shifts the Disjoint Blow-up/Collapse property and its strong version are equivalent.Our main result is that if  1 , . . .,   satisfy the Strong Disjoint Blowup/Collapse property, then they have a dense linear manifold of disjoint hypercyclic vectors.We conclude the paper with some open questions.

Preliminary Results
For convenience, in all what follows, the open neighbourhoods of zero will be chosen to be balanced; that is,  = ∪ ||≤1 .Definition 1.The operators  1 , . . .,   ∈ B() are said to satisfy the Strong Disjoint Blow-up/Collapse property if for all  ≥ 1 whenever nonempty open sets ,  1− , . . .,  0 ,  1 , . . .,   are given with  a neighborhood of 0, then there exists  such that Remark 2. (a) The Disjoint Blow-up/Collapse property, Definition E, results when  is only allowed to be 1.(b) If  satisfies the Blow-up/Collapse property, then by using that  satisfies the Hypercyclicity Criterion (see paragraph after Definition B) one can prove that  also satisfies the Strong Blow-up/Collapse property.Proposition 2.4 of [11], which was stated without proof, results from the following proposition when  =  0 ,  = 1, and the word "strong" is omitted. Let As indicated in the introduction, it was pointed out in [18] that the following proposition is true when the word "strong" is eliminated.Proof.Let  be an open neighbourhood of zero and  1− , . . .,  0 ,  1 , . . .,   ⊂  be nonempty open sets.Assume that  0 ,  1 , . . .,   are the dense subsets of  and  , :   →  with 1 ≤  ≤  the mappings given by the Disjoint Hypercyclic Criterion.Let   ∈   ∩  0 for 1 −  ≤  ≤ 0 and   ∈   ∩   for 1 ≤  ≤ .We now choose another open neighbourhood of zero   with   ⊂  and such that 1 ≤  ≤  and By hypothesis there exists   so that     (  ) ∈   , for all 1 ≤  ≤  and 1 −  ≤  ≤ 0, and, for 1 ≤  ≤ ,  , (  ) ∈   , Thus ( 5) is satisfied.It remains to verify that ( 6) is also satisfied.For that we choose  0 = ∑  =1  , (  ) and it follows that The following proposition has an immediate proof, and it is often used when studying weighted shifts.Let K be either N or Z, and let  be   (K) for 1 ≤  < ∞ or  0 (K).Let {  :  ∈ K} be the canonical basis of  and for  ∈  we denote by ⟨, The following result says that for the class of -tuples of weighted shifts the Disjoint Blow-up/Collapse property and its strong version coincide.Proposition 6.Let  be   (K) for 1 ≤  < ∞ or  0 (K).If  1 , . . .,   are disjoint hypercyclic weighted shifts, then they satisfy the Strong Disjoint Blow-up/Collapse property.
Proof.By Theorems 2.1 and 2.2 of [18] these operators satisfy the Disjoint Blow-up/Collapse property.We have to prove that this implies that the strong version is also satisfied.
In the following section we give a partial affirmative answer whenever  1 , . . .,   satisfy the Strong Disjoint Blowup/Collapse property.

Main Result
In the theorem below, it is worth noting that it is not necessary to suppose beforehand that the operators have a dense set of disjoint hypercyclic vectors; this follows from the construction.However, since  1 , . . .,   ∈ B() satisfy the Disjoint Blow-up/Collapse property, they have a dense set of disjoint hypercyclic vectors which is a   , Proposition 2.3 of [11].
Theorem 8. Let  be a topological vector space, over either R or C, whose topology has a countable basis and is complete.Let  1 , . . .,   ∈ B() satisfy the Strong Disjoint Blowup/Collapse property.Then  1 , . . .,   have a dense linear manifold of disjoint hypercyclic vectors.
Proof.We prove the theorem when  = 2.The proof for an arbitrary  is conceptually the same, but the notation is more cumbersome.
The setting up of the proof is as follows.For each  we find a sequence {(, , )} with 1 ≤  ≤  +  and such that the order in which the vectors are generated is the lexicographic order for (, ).The limit for  → ∞ of (, , ) =   will exist.The linear manifold of disjoint hypercyclic vectors is the span of the    .Let {( ,1 ,  ,2 ) :  ∈ N,  = 1, 2} be dense in  × , and let {  :  ∈ N} be dense in .Let  1 ∪  2 ∪  3 ∪ ⋅ ⋅ ⋅ = N be a partition of N such that each   is an infinite set and define   =   whenever  ∈   .
Let {  :  ∈ N} be a local basis of 0 such that each   is a balanced open set and  +1 ⊊   .
We now proceed to the construction of the vectors {(, , )}.In each step after the third one, we find several vectors at the same time thanks to the strong version of the Disjoint Blow-up/Collapse property, Definition 1. (We use properties ( 5) and ( 7).) Step  corresponds to the (, ) for which  = (+−1)(+ )/2 + 1 − .In this step  +  − 1 vectors are found.
The authors of [6] comment that they do not know the answer to Problem 3.6 of [6] (Problem G) even when the operators are weighted shifts.
Proof.Proposition 6 and Theorem 8 provide the proof.

Concluding Questions
Among the following questions the most fundamental is the first one.The next three questions might be easier to handle for the class of -tuples of weighted shifts thanks to the results of [18] which also characterize disjoint hypercyclicity in terms of their weight sequence.
First we recall the relevant definitions.A hypercyclic subspace for  ∈ B() is an infinite dimensional subspace whose nonzero vectors are hypercyclic.The systematic study of hypercyclic subspaces started with work by Bernal-González and Montes-Rodríguez in [21].Chapter 8 of [8] and Chapter 10 of [9] give the fundamentals and history of hypercyclic subspaces.
The operator  ∈ B() is supercyclic if there is an  ∈  such that {   :  ∈ N,  scalar} is dense in .Chapter 9 of [8] treats supercyclicity.A supercyclic subspace for  ∈ B() is an infinite dimensional subspace whose nonzero vectors are supercyclic.A panorama of supercyclic subspaces is given by Montes-Rodríguez and Salas in their survey [22].