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 .
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  and by Gethner and Shapiro in . 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 . Bernal-González and Grosse-Erdmann  and León-Saavedra  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  showed that this is not always the case.
The excellent books by Bayart and Matheron  and Grosse-Erdmann and Peris  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 . The following concept was introduced independently by Bernal-González  and Bès and Peris .
Definition C. Let . The operators are disjoint hypercyclic if there is an such that the orbit is dense in . The vector is called a disjoint hypercyclic vector for .
It is worth noting that while the author of  was inspired by some recent work by Costakis and Vlachou in universal Taylor series, the authors of  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  in a slightly more general way. The first one is their Definition 2.1.
Definition D. Let . The operators are disjoint topologically transitive if whenever are nonempty open sets, then there exists such that
The second one is included in their Proposition 2.4.
Definition E. Let . The operators satisfy the Disjoint Blow-up/Collapse property if for each open neighbourhood of zero and nonempty open subsets there exists such that
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 . Let be a strictly increasing sequence of positive integers. The operators satisfy the Disjoint Hypercyclicity Criterion with respect to provided there exist dense subsets of and mappings with and 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  says that are disjoint topologically transitive if and only if the set of disjoint hypercyclic vectors for is a dense of .
Proposition 2.4 of  says that if satisfy the Disjoint Blow-up/Collapse property, then they are disjoint topologically transitive. This is the disjoint version of Theorem of , but the authors of  assume that is a Banach space.
Theorem of  says that satisfy the Disjoint Hypercyclicity Criterion if and only if for each the operators are disjoint topologically transitive in .
Two comments are in order.(a)Theorem 2.7 of  also gives another equivalence.(b)When , Theorem 2.7 of  is another equivalence of the Hypercyclicity Criterion. This equivalence is given by the same authors in an earlier paper, Theorem 2.3 of .
Assume . Bès et al. show in Theorems 2.1 and 2.2 of  that weighted shifts are disjoint hypercyclic if and only if they satisfy the Disjoint Blow-up/Collapse property. They also show that weighted shifts can never satisfy the Disjoint Hypercyclicity Criterion; see Proposition 3.2 in . The relative simplicity with which the authors of  show disjoint hypercyclic operators which do not satisfy the Disjoint Hypercyclicity Criterion should be contrasted with the sophistication of the arguments in .
The authors of  also point out that if satisfy the Disjoint Hypercyclicity Criterion, then satisfy the Disjoint Blow-up/Collapse property; see their last diagram.
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 satisfy the Strong Disjoint Blow-up/Collapse property, then they have a dense linear manifold of disjoint hypercyclic vectors. We conclude the paper with some open questions.
2. Preliminary Results
For convenience, in all what follows, the open neighbourhoods of zero will be chosen to be balanced; that is, .
Definition 1. The operators are said to satisfy the Strong Disjoint Blow-up/Collapse property if for all whenever nonempty open sets 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 , which was stated without proof, results from the following proposition when , , and the word “strong’’ is omitted.
Proposition 3. Let satisfy the Strong Disjoint Blow-up/Collapse property. If and the nonempty open sets and are given with a neighborhood of 0, then there exists such that (5) holds and
Proof. Let for and . Let be an open set containing 0 such that and for and .
Set and for and for . Since satisfy the Strong Disjoint Blow-up/Collapse property, we have that there exist and for and such that and for and . It remains to check that Let and for . Then .
As indicated in the introduction, it was pointed out in  that the following proposition is true when the word “strong’’ is eliminated.
Proposition 4. If satisfy the Disjoint Hypercyclicity Criterion, then they also satisfy the Strong Disjoint Blow-up/Collapse property.
Proof. Let be an open neighbourhood of zero and be nonempty open sets. Assume that are the dense subsets of and with the mappings given by the Disjoint Hypercyclic Criterion. Let for and for . We now choose another open neighbourhood of zero with and such that and
By hypothesis there exists so that , for all and , and, for ,
Thus (5) is satisfied. It remains to verify that (6) is also satisfied. For that we choose and it follows that
The following proposition has an immediate proof, and it is often used when studying weighted shifts. Let be either or , and let be for or . Let be the canonical basis of and for we denote by , where and . We say that the vector dominates the vector if
Proposition 5. Let be a weighted shift. If the vector dominates the vector , then for all ,
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 for or . If are disjoint hypercyclic weighted shifts, then they satisfy the Strong Disjoint Blow-up/Collapse property.
Proof. By Theorems 2.1 and 2.2 of  these operators satisfy the Disjoint Blow-up/Collapse property. We have to prove that this implies that the strong version is also satisfied.
We prove it for and for or which illustrate the general method. Let and for some . Let and be the orthonormal canonical basis, respectively, with respect to which both operators are weighted shifts. We can assume without loss of generality that there exist in the span of such that for and is a finite interval of either or and . Let . Let us choose such that if and 0 otherwise. Set . Apply the Disjoint Blow-up/Collapse property to and , , and . This means that there is an arbitrarily large such that
Case 1 . If is large enough, we have that for all in the span of , in particular for . Thus for .
Case 2 . There exists such that and . Since and , we have for the following: for ; but for . Thus are dominated by , and we are done.
Corollary 7. The converse of Proposition 4 is not true.
Proof. By Theorems 2.1 and 2.2 of  disjoint hypercyclic weighted shifts satisfy the Disjoint Blow-up/Collapse property but by Proposition 2.3 of  cannot satisfy the Disjoint Hypercyclicity Criterion. By using Proposition 6 we can conclude the proof of the corollary.
If is hypercylic, Herrero  and Bourdon , independently, showed that has a dense linear manifold of hypercyclic vectors. (See also page 53 of .) If are disjoint hypercyclic, it is not known whether their set of disjoint hypercyclic vectors is dense in , page 115 of . In view of the above results, the authors of  pose their Problem 3.6 which is the following.
Problem G. Let be densely disjoint hypercyclic operators in . Must they support a dense disjoint hypercyclic manifold?
In the following section we give a partial affirmative answer whenever satisfy the Strong Disjoint Blow-up/Collapse property.
3. 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 satisfy the Disjoint Blow-up/Collapse property, they have a dense set of disjoint hypercyclic vectors which is a , Proposition 2.3 of .
Theorem 8. Let be a topological vector space, over either or , whose topology has a countable basis and is complete. Let satisfy the Strong Disjoint Blow-up/Collapse property. Then have a dense linear manifold of disjoint hypercyclic vectors.
Proof. We prove the theorem when . 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 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 be dense in , and let be dense in . Let be a partition of such that each is an infinite set and define whenever .
Let be a local basis of 0 such that each is a balanced open set and .
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 . In this step vectors are found.
We choose for all , and, moreover, for all and . We also have that their limit . In this way we ensure that is dense in since for we have that .
Step 1. Let and for . Then by (7) we have and such that for
Since , are continuous, there exists such that and implies that for . In addition, since is a local basis, we can choose sufficiently small that .
Step 2. Let and for . Let and . By applying (5) and (7) we find and and such that for
Let be such that and implies that , and if , then . Also we can choose such that .
Step n. Let be such that .
The open set has been chosen in the previous step. Let
Case We have that since . Set and for let Case Set and for let
Case Set and for let and for let
Setting and using (5) and (7) there are vectors and such that satisfy, for , And the remaining vectors chosen in this step satisfy the following displayed formulas. (Clearly when only (28) matters, whereas when only (29) matters.)
For we have that
For we have and
We now choose a sufficiently small open set with and such that implies that and for the other vectors with chosen at this stage we also have that implies that for . Moreover, we also have that for all vectors chosen at this stage where is the immediate predecessor of in the lexicographic order. When we have that
Since is also a local basis of 0 and is complete, it follows from (29) and (30) that for each the sequence converges to a vector . Moreover, (27) and (28) and imply that for .
To finish the proof we have to prove that if with some , then is disjoint hypercyclic. To that end choose such that for . Since nonzero multiples of disjoint hypercyclic vectors are also disjoint hypercyclic, we may assume that . For and we have that and for and we have that , where the vector is obtained in the step. Therefore for we use (31) to get since for and the open sets have been chosen to be balanced. Thus we have shown that is a disjoint hypercyclic vector.
Corollary 9. Let be for or . If are disjoint hypercyclic weighted shifts, then they have a dense linear manifold of disjoint hypercyclic vectors.
4. 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  which also characterize disjoint hypercyclicity in terms of their weight sequence.
First we recall the relevant definitions. A hypercyclic subspace for 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 . Chapter 8 of  and Chapter 10 of  give the fundamentals and history of hypercyclic subspaces.
The operator is supercyclic if there is an such that is dense in . Chapter 9 of  treats supercyclicity. A supercyclic subspace for 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 .
A disjoint hypercyclic subspace for is an infinite dimensional subspace whose nonzero vectors are disjoint hypercyclic. Proposition 3.7 of  assures the existence of disjoint hypercyclic subspaces in some cases. Another line of inquiry is to study disjoint supercyclicity; see [14–16]. Montes-Rodríguez and Salas characterized supercyclic subspaces for the class of weighted shifts ; see also .
Question 1. Are the Disjoint Blow-up/Collapse property and its strong version equivalent?
Question 2. Given that satisfy the Disjoint Blow-up/Collapse property, can we add some such that the resultant -tuple still satisfies the Disjoint Blow-up/Collapse property?
Question 3. Given which are disjoint hypercyclic, can we add some such that the resultant -tuple is still disjoint hypercyclic?
Question 4. Assume that have a disjoint hypercyclic space. Can we add some such that the resultant -tuple still has a disjoint hypercyclic subspace?
In the last three questions is either for or .
Question 5. Which weighted shifts have a disjoint hypercyclic space?
Question 6. Which weighted shifts are disjoint supercyclic?
Question 7. Which weighted shifts have a disjoint supercyclic space?
More unsolved questions on disjoint hypercyclicity can be found in [10–12, 14, 15, 18]. Unsolved questions in supercyclicity can be found in [22, 23].
The author would like to thank the referees for their very useful comments. I would also like to thank Professor Dorothy Bollman for her invaluable insights.
J. H. Shapiro, “Notes on the dynamics of linear operators,” http://www.math.msu.edu/~shapiro/Pubvit/Downloads/LinDynamics/lindynamics.pdf.View at: Google Scholar
C. Kitai, Invariant closed sets for linear operators [Ph.D. thesis], University of Toronto, 1982.
J. Bès, Ö. Martin, and R. Sanders, “Weighted shifts and disjoint hypercyclicity,” submitted to Journal of Operator Theory. In press.View at: Google Scholar