The Strong Disjoint Blow-Up/Collapse Property

Abstract

Let 𝑋 be a topological vector space, and let ℬ(𝑋) be the algebra of continuous linear operators on 𝑋. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are disjoint hypercyclic if there is 𝑥∈𝑋 such that the orbit {(𝑇^𝑛₁(𝑥),…,𝑇^𝑛_𝑁(𝑥))∶𝑛∈ℕ} is dense in 𝑋×…×𝑋. Bès and Peris have shown that if 𝑇₁,…,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 𝑇₁,…,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.

1. Introduction and Background

Let 𝑋 be a topological vector space, over either the real or complex numbers, whose topology has a countable basis and is complete. Let ℬ(𝑋) be the algebra of continuous linear operators on 𝑋.

The operator 𝑇∈ℬ(𝑋) is hypercyclic if there is 𝑥∈𝑋 such that Orb(𝑇,𝑥)∶={𝑇^𝑘𝑥∶𝑘∈ℕ} is dense in 𝑋. This concept is closely related to the concept of transitivity from topological dynamics.

Definition A. The operator 𝑇∈ℬ(𝑋) is topologically transitive if for each pair 𝑈, 𝑉 of nonempty open subsets of 𝑋 there is 𝑛 such that 𝑈∩𝑇^−𝑛(𝑉)≠∅.

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 𝑇∈ℬ(𝑋) satisfies the Blow-up/Collapse property if whenever nonempty open sets 𝑊, 𝑉₀, 𝑉₁ are given with 𝑊 neighbourhood of 0, then there exits 𝑛 such that 𝑇^−𝑛(𝑊)∩𝑉₀≠∅,𝑇^−𝑛𝑉₁∩𝑊≠∅. (1)

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 𝑇 satisfies the Blow-up/Collapse property if and only if 𝑇 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].

Definition C. Let 𝑁≥2. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are disjoint hypercyclic if there is an 𝑥∈𝑋 such that the orbit {(𝑇^𝑛₁(𝑥),…,𝑇^𝑛_𝑁(𝑥))∶𝑛∈ℕ} is dense in 𝑋×⋯×𝑋. The vector 𝑥 is called a disjoint hypercyclic vector for 𝑇₁,…,𝑇_𝑁.

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 papers by Salas [12], Shkarin [13], Bès et al. [14, 15], and Bès and Martin [16] further explore different aspects of disjoint hypercyclicity.

The following three definitions were given in [11] in a slightly more general way. The first one is their Definition 2.1.

Definition D. Let 𝑁≥2. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are disjoint topologically transitive if whenever 𝑉₀,𝑉₁,…,𝑉_𝑁⊂𝑋 are nonempty open sets, then there exists 𝑛∈ℕ such that 𝑇₁^−𝑛𝑉₁∩⋯∩𝑇_𝑁^−𝑛𝑉_𝑛∩𝑉₀≠∅. (2)

The second one is included in their Proposition  2.4.

Definition E. Let 𝑁≥2. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Blow-up/Collapse property if for each open neighbourhood of zero 𝑊 and nonempty open subsets 𝑉₀,𝑉₁,…,𝑉_𝑁⊂𝑋 there exists 𝑛∈ℕ such that 𝑇₁^−𝑛(𝑊)∩⋯∩𝑇_𝑁^−𝑛(𝑊)∩𝑉₀𝑇≠∅,_1−𝑛𝑉₁∩⋯∩𝑇_𝑁^−𝑛𝑉_𝑁∩𝑊≠∅. (3)

The third one is their Definition 2.5. (If 𝑁 were allowed to be 1, then this would be one of the many versions of the classical Hypercyclicity Criterion.)

Definition F. Let 𝑁≥2. Let (𝑛_𝑘) be a strictly increasing sequence of positive integers. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Hypercyclicity Criterion with respect to (𝑛_𝑘) provided there exist dense subsets 𝑋₀,𝑋₁,…,𝑋_𝑁 of 𝑋 and mappings 𝑆_𝑙,𝑘∶𝑋_𝑙→𝑋 with 1≤𝑙≤𝑁 and 𝑘∈ℕ satisfying 𝑇^𝑛𝑘_𝑙⟶0pointwiseon𝑋₀,𝑆_𝑙,𝑘⟶0pointwiseon𝑋_𝑙,𝑇^𝑛𝑘_𝑙𝑆_𝑖,𝑘−𝛿_𝑖,𝑙𝐼𝑑_𝑋𝑖⟶0pointwiseon𝑋_𝑖with1≤𝑖≤𝑁. (4)

In the last display 𝐼𝑑_𝑋𝑖 means the identity in 𝑋_𝑖. (It has to be 𝑋_𝑖 and not 𝑋_𝑙 because 𝑋_𝑖 is the domain of 𝑆_𝑖,𝑘.) We now give some known relations between all these concepts.

Proposition 2.3 of [11] says that 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are disjoint topologically transitive if and only if the set of disjoint hypercyclic vectors for 𝑇₁,…,𝑇_𝑁 is a dense 𝐺_𝛿 of 𝑋.

Proposition 2.4 of [11] says that if 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) 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 𝑋 is a Banach space.

Theorem 2.7 of [11] says that 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Hypercyclicity Criterion if and only if for each 𝑟∈ℕ the operators 𝑇^𝑟₁,…,𝑇₁,…,𝑇_𝑁,…^𝑟,𝑇_𝑁 are disjoint topologically transitive in ℬ(𝑋^𝑟).

Two comments are in order.(a)Theorem 2.7 of [11] also gives another equivalence.(b)When 𝑁=1, Theorem 2.7 of [11] is another equivalence of the Hypercyclicity Criterion. This equivalence is given by the same authors in an earlier paper, Theorem 2.3 of [17].

Assume 2≤𝑁. Bès et al. show in Theorems 2.1 and 2.2 of [18] that 𝑁 weighted shifts are disjoint hypercyclic if and only if they satisfy the Disjoint Blow-up/Collapse property. They also show that 𝑁 weighted shifts can never satisfy the Disjoint Hypercyclicity Criterion; see Proposition 3.2 in [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].

The authors of [18] also point out that if 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Hypercyclicity Criterion, then 𝑇₁,…,𝑇_𝑁 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 𝑁-tuples of weighted shifts the Disjoint Blow-up/Collapse property and its strong version are equivalent. Our main result is that if 𝑇₁,…,𝑇_𝑁 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, 𝑊=∪_|𝜆|≤1𝜆𝑊.

Definition 1. The operators 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are said to satisfy the Strong Disjoint Blow-up/Collapse property if for all 𝑚≥1 whenever nonempty open sets 𝑊,𝑉_1−𝑚,…,𝑉₀,𝑉₁,…,𝑉_𝑁 are given with 𝑊 a neighborhood of 0, then there exists 𝑛 such that 𝑇₁^−𝑛(𝑊)∩⋯∩𝑇_𝑁^−𝑛(𝑊)∩𝑉_𝑘−𝑚𝑇≠∅for1≤𝑘≤𝑚,_1−𝑛𝑉₁∩⋯∩𝑇_𝑁^−𝑛𝑉_𝑁∩𝑊≠∅. (5)(6)

Remark 2. (a) The Disjoint Blow-up/Collapse property, Definition E, results when 𝑚 is only allowed to be 1.
(b) If 𝑇 satisfies the Blow-up/Collapse property, then by using that 𝑇 satisfies the Hypercyclicity Criterion (see paragraph after Definition B) one can prove that 𝑇 also satisfies the Strong Blow-up/Collapse property.

Proposition 2.4 of [11], which was stated without proof, results from the following proposition when 𝑈=𝑉₀, 𝑚=1, and the word “strong’’ is omitted.

Proposition 3. Let 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Strong Disjoint Blow-up/Collapse property. If 𝑚≥1 and the nonempty open sets 𝑊,𝑈,𝑉₁,…,𝑉_𝑁 and 𝑉_1−𝑚,…,𝑉₀ are given with 𝑊 a neighborhood of 0, then there exists 𝑛 such that (5) holds and 𝑇₁^−𝑛𝑉₁∩⋯∩𝑇_𝑁^−𝑛𝑉_𝑁∩𝑈≠∅. (7)

Proof. Let 𝑥_𝑗∈𝑉_𝑗 for 1≤𝑗≤𝑁 and 𝑥∈𝑈. Let 𝑊^′ be an open set containing 0 such that 𝑊^′+𝑊^′⊂𝑊 and 𝑥_𝑗+𝑊^′+𝑊^′⊂𝑉_𝑗 for 1≤𝑗≤𝑁 and 𝑥+𝑊^′+𝑊^′⊂𝑈.
Set 𝑉^′_−𝑚=𝑥+𝑊^′ and 𝑉^′_𝑖=𝑉_𝑖 for 1−𝑚≤𝑖≤0 and 𝑉^′_𝑗=𝑥_𝑗+𝑊^′ for 1≤𝑗≤𝑁. Since 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Strong Disjoint Blow-up/Collapse property, we have that there exist 𝑛∈ℕ and 𝑧_𝑖∈𝑉^′_𝑖 for −𝑚≤𝑖≤0 and 𝑤∈𝑊^′ such that 𝑇^𝑛_𝑗(𝑧_𝑖)∈𝑊^′ and 𝑇^𝑛_𝑗(𝑤)∈𝑉^′_𝑗 for −𝑚≤𝑖≤0 and 1≤𝑗≤𝑁. It remains to check that 𝑇₁^−𝑛𝑉₁∩⋯∩𝑇_𝑁^−𝑛𝑉_𝑁∩𝑈≠∅. (8) Let 𝑧=𝑧_−𝑚+𝑤∈𝑥+𝑊^′+𝑊^′⊂𝑈 and 𝑇^𝑛_𝑗(𝑧)∈𝑉^′_𝑗+𝑊^′⊂𝑉_𝑗 for 1≤𝑗≤𝑁. Then 𝑧∈𝑇₁^−𝑛(𝑉₁)∩⋯∩𝑇_𝑁^−𝑛(𝑉_𝑁)∩𝑈.

As indicated in the introduction, it was pointed out in [18] that the following proposition is true when the word “strong’’ is eliminated.

Proposition 4. If 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Hypercyclicity Criterion, then they also satisfy the Strong Disjoint Blow-up/Collapse property.

Proof. Let 𝑊 be an open neighbourhood of zero and 𝑉_1−𝑚,…,𝑉₀,𝑉₁,…,𝑉_𝑁⊂𝑋 be nonempty open sets. Assume that 𝑋₀,𝑋₁,…,𝑋_𝑁 are the dense subsets of 𝑋 and 𝑆_𝑙,𝑘∶𝑋_𝑙→𝑋 with 1≤𝑙≤𝑁 the mappings given by the Disjoint Hypercyclic Criterion. Let 𝑧_𝑗∈𝑉_𝑗∩𝑋₀ for 1−𝑚≤𝑗≤0 and 𝑦_𝑖∈𝑉_𝑖∩𝑋_𝑖 for 1≤𝑖≤𝑁. We now choose another open neighbourhood of zero 𝑊^′ with 𝑊^′⊂𝑊 and such that 1≤𝑖≤𝑁 and 𝑊^𝑁′+⋯+𝑊^′⊂𝑊,𝑦_𝑖+𝑊^′+⋯^𝑁+𝑊^′⊂𝑉_𝑖. (9) By hypothesis there exists 𝑛_𝑘 so that 𝑇^𝑛𝑘_𝑖(𝑧_𝑗)∈𝑊^′, for all 1≤𝑖≤𝑁 and 1−𝑚≤𝑗≤0, and, for 1≤𝑙≤𝑁, 𝑆_𝑙,𝑘𝑦_𝑙∈𝑊^′,𝑇^𝑛𝑘_𝑙𝑆_𝑖,𝑘𝑦_𝑖−𝛿_𝑙,𝑖𝑦_𝑖⊂𝑊^′. (10)
Thus (5) is satisfied. It remains to verify that (6) is also satisfied. For that we choose 𝑤₀=∑^𝑁_𝑖=1𝑆_𝑖,𝑘(𝑦_𝑖) and it follows that 𝑤₀∈𝑇^−𝑛𝑘₁𝑉₁∩⋯∩𝑇^−𝑛𝑘_𝑁𝑉_𝑁∩𝑊. (11)

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 𝑋 be 𝑙^𝑝(𝕂) for 1≤𝑝<∞ or 𝑐₀(𝕂). Let {𝑒_𝑖∶𝑖∈𝕂} be the canonical basis of 𝑋 and for 𝑥∈𝑋 we denote by ⟨𝑥,𝑒^∗_𝑖⟩=𝑒^∗_𝑖(𝑥), where 𝑒^∗_𝑖∈𝑋^∗ and 𝑒^∗_𝑖(𝑒_𝑗)=𝛿_𝑖,𝑗. We say that the vector 𝑦  dominates the vector 𝑥 if 𝑥,𝑒_𝑖≤𝑦,𝑒_𝑖∀𝑖. (12)

Proposition 5. Let 𝑇 be a weighted shift. If the vector 𝑎 dominates the vector 𝑏, then for all 𝑛∈ℕ, 𝑇^𝑛(𝑏)≤𝑇^𝑛(𝑎). (13)

The following result says that for the class of 𝑁-tuples of weighted shifts the Disjoint Blow-up/Collapse property and its strong version coincide.

Proposition 6. Let 𝑋 be 𝑙^𝑝(𝕂) for 1≤𝑝<∞ or 𝑐₀(𝕂). If 𝑇₁,…,𝑇_𝑁 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.
We prove it for 𝑁=2 and for 𝑙²(ℕ) or 𝑙²(ℤ) which illustrate the general method. Let 𝑊={𝑥∶‖𝑥‖<𝛿} and 𝑉₂,𝑉₁,𝑉₀,…,𝑉_−𝑘 for some 𝑘∈ℕ. Let {𝑒_𝑖∶𝑖∈ℕ} and {𝑒_𝑖∶𝑖∈ℤ} 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 𝑥₀,…,𝑥_−𝑘 in the span of {𝑒_𝑖∶𝑖∈𝐴} such that {𝑥∶‖𝑥−𝑥_𝑗‖<𝜖}⊂𝑉_−𝑗 for 𝑗=0,…,𝑘 and 𝐴 is a finite interval of either ℕ or ℤ and 𝜖<1/2. Let 𝑀=max{||𝑥_−𝑗||+1∶𝑗=0,…,𝑘}. Let us choose 𝑧₀ such that ⟨𝑧,𝑒_𝑖⟩=𝑀 if 𝑖∈𝐴 and 0 otherwise. Set 𝑉₀={𝑥∶‖𝑥−𝑧‖<𝜖}. Apply the Disjoint Blow-up/Collapse property to 𝑊 and 𝑉₂, 𝑉₁, and 𝑉₀. This means that there is an arbitrarily large 𝑛 such that 𝑉₀∩𝑇₁^−𝑛(𝑊)∩𝑇₂^−𝑛(𝑊)≠∅,𝑊∩𝑇₁^−𝑛𝑉₁∩𝑇₂^−𝑛𝑉₂≠∅. (14)
Case  1  (𝑙²(ℕ)). If 𝑛 is large enough, we have that 𝑇^𝑛₁(𝑥)=0=𝑇^𝑛₂(𝑥) for all 𝑥 in the span of {𝑒_𝑖∶𝑖∈𝐴}, in particular for 𝑥₀,…,𝑥_−𝑘. Thus 𝑉_−𝑗∩𝑇₁^−𝑛(𝑊)∩𝑇₂^−𝑛(𝑊)≠∅ (15) for 𝑗=0,…,𝑘.
Case  2  (𝑙²(ℤ)). There exists 𝑧∈𝑉₀ such that ‖𝑇^𝑛₁(𝑧)‖<𝛿 and ‖𝑇^𝑛₂(𝑧)‖<𝛿. Since 𝜖<1/2 and ‖𝑒_𝑖‖=1, we have for 𝑗=0,…,𝑘 the following: 𝑧,𝑒_𝑖≥𝑧₀,𝑒_𝑖−𝑧−𝑧₀1≥𝑀−2>𝑥_−𝑗≥𝑥_−𝑗,𝑒_𝑖 (16) for 𝑖∈𝐴; but for 𝑖∉𝐴,|⟨𝑧,𝑒_𝑖⟩|≥0=|⟨𝑥_−𝑗,𝑒_𝑖⟩|. Thus 𝑥₀,…,𝑥_−𝑘 are dominated by 𝑧, and we are done.

Corollary 7. The converse of Proposition 4 is not true.

Proof. By Theorems 2.1 and 2.2 of [18] disjoint hypercyclic weighted shifts satisfy the Disjoint Blow-up/Collapse property but by Proposition 2.3 of [18] cannot satisfy the Disjoint Hypercyclicity Criterion. By using Proposition 6 we can conclude the proof of the corollary.

If 𝑇∈ℬ(𝑋) is hypercylic, Herrero [19] and Bourdon [20], independently, showed that 𝑇 has a dense linear manifold of hypercyclic vectors. (See also page 53 of [9].) If 𝑇₁,…,𝑇_𝑁 are disjoint hypercyclic, it is not known whether their set of disjoint hypercyclic vectors is dense in 𝑋, page 115 of [10]. In view of the above results, the authors of [18] pose their Problem 3.6 which is the following.

Problem G.  Let 𝑇₁,…,𝑇_𝑁 be densely disjoint hypercyclic operators in ℬ(𝑋). Must they support a dense disjoint hypercyclic manifold?

In the following section we give a partial affirmative answer whenever 𝑇₁,…,𝑇_𝑁 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 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Disjoint Blow-up/Collapse property, they have a dense set of disjoint hypercyclic vectors which is a 𝐺_𝛿, Proposition 2.3 of [11].

Theorem 8. Let 𝑋 be a topological vector space, over either ℝ or ℂ, whose topology has a countable basis and is complete. Let 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) satisfy the Strong Disjoint Blow-up/Collapse property. Then 𝑇₁,…,𝑇_𝑁 have a dense linear manifold of disjoint hypercyclic vectors.

Proof. We prove the theorem when 𝑁=2. The proof for an arbitrary 𝑁 is conceptually the same, but the notation is more cumbersome.
The setting up of the proof is as follows. For each 𝑖 we find a sequence {𝑥(𝑖,𝑘,𝑗)} with 1≤𝑗≤𝑖+𝑘 and such that the order in which the vectors are generated is the lexicographic order for (𝑘,𝑗). The limit for 𝑘→∞ of 𝑥(𝑖,𝑘,𝑗)=𝑥_𝑖 will exist. The linear manifold of disjoint hypercyclic vectors is the span of the 𝑥^′_𝑖𝑠.
Let {(𝑦_𝑘,1,𝑦_𝑘,2)∶𝑘∈ℕ,𝑢=1,2} be dense in 𝑋×𝑋, and let {𝑧_𝑝∶𝑝∈ℕ} be dense in 𝑋. Let 𝐴₁∪𝐴₂∪𝐴₃∪⋯=ℕ be a partition of ℕ such that each 𝐴_𝑝 is an infinite set and define 𝑓_𝑗=𝑧_𝑝 whenever 𝑗∈𝐴_𝑝.
Let {𝐺_𝑛∶𝑛∈ℕ} be a local basis of 0 such that each 𝐺_𝑛 is a balanced open set and 𝐺_𝑛+1⊊𝐺_𝑛.
We now proceed to the construction of the vectors {𝑥(𝑖,𝑘,𝑗)}. 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 𝑛 corresponds to the (𝑖,𝑘) for which 𝑛=(𝑖+𝑘−1)(𝑖+𝑘)/2+1−𝑖. In this step 𝑖+𝑘−1 vectors are found.
We choose 𝑥(𝑖,1,1)∈𝑓_𝑖+𝐺_𝑖 for all 𝑖∈ℕ, and, moreover, 𝑥(𝑖,𝑘,𝑗)∈𝑓_𝑖+𝐺_𝑖 for all 𝑘 and 1≤𝑗≤𝑖+𝑘. We also have that their limit 𝑥_𝑖∈𝑓_𝑖+𝐺_𝑖. In this way we ensure that {𝑥_𝑖∶𝑖∈ℕ} is dense in 𝑋 since for 𝑗∈𝐴_𝑝 we have that lim_𝑗→∞𝑥_𝑗=𝑧_𝑝.
Step  1. Let 𝑈=𝑓₁+𝐺₁ and 𝑉_𝑢=𝑦_1,𝑢+𝐺₁ for 𝑢=1,2. Then by (7) we have 𝑥(1,1,1)∈𝑈 and 𝑚₁ such that for 𝑢=1,2𝑇^𝑚1_𝑢(𝑥(1,1,1))∈𝑉_𝑢. (17)
Since 𝑇₁, 𝑇₂ are continuous, there exists 𝑊₂=𝐺_𝑝2 such that 1=𝑝₁<𝑝₂ and 𝑥−𝑥(1,1,1)∈𝑊₂ implies that 𝑇^𝑚1_𝑢(𝑥)∈𝑉_𝑢 for 𝑢=1,2. In addition, since {𝐺_𝑘} is a local basis, we can choose 𝑊₂ sufficiently small that 𝑥(1,1,1)+𝑊₂⊂𝑓₁+𝐺₁.
Step  2. Let 𝑈=𝑓₂+𝐺₂ and 𝑉_𝑢=𝑦_2,𝑢+𝐺₂ for 𝑢=1,2. Let 𝑉₀=𝑥(1,1,1)+𝑊₂ and 𝑊=𝑊₂. By applying (5) and (7) we find 𝑥(2,1,1)∈𝑈 and 𝑥(1,1,2)∈𝑉₀ and 𝑚₂ such that for 𝑢=1,2𝑇^𝑚2_𝑢(𝑥(2,1,1))∈𝑉_𝑢,𝑇^𝑚2_𝑢(𝑥(1,1,2))∈𝑊₂. (18)
Let 𝑊₃=𝐺_𝑝3 be such that 𝑝₂<𝑝₃ and 𝑥−𝑥(2,1,1)∈𝑊₃ implies that 𝑇^𝑚2_𝑢(𝑥)−𝑦_2,𝑢∈𝑊₂, and if 𝑥−𝑥(1,1,2)∈𝑊₃, then 𝑇^𝑚2_𝑢(𝑥)∈𝑊₂. Also we can choose 𝑊₃ such that 𝑥(1,1,2)+𝑊₃⊂𝑉₀.
Step  n. Let (𝑖,𝑘) be such that 𝑛=(𝑖+𝑘−1)(𝑖+𝑘)/2+1−𝑖.
The open set 𝑊_𝑛=𝐺_𝑝𝑛 has been chosen in the previous step. Let 𝑉_𝑢=𝑦_𝑛,𝑢+𝐺_𝑛for𝑢=1,2. (19)
Case  (𝑘=1).We have that 1<𝑖 since 3≤𝑛. Set 𝑈=𝑓_𝑖+𝐺_𝑖 and for 0≤𝑡≤𝑖−2 let 𝑥(𝑖−1−𝑡,1+𝑡,𝑖−1−𝑡)+𝑊_𝑛=𝑉_−𝑡. (20)Case  (𝑖=1).Set 𝑈=𝑥(1,𝑘−1,𝑘)+𝑊_𝑛 and for 2≤𝑠≤𝑘 let 𝑥(𝑠,𝑘+1−𝑠,𝑠−1)+𝑊_𝑛=𝑉_2−𝑠. (21)
Case  (1<𝑘,1<𝑖).Set 𝑈=𝑥(𝑖,𝑘−1,𝑖+𝑘−1)+𝑊_𝑛 and for 0≤𝑡≤𝑖−2 let 𝑥(𝑖−1−𝑡,𝑘+𝑡,𝑘+𝑖−2−𝑡)+𝑊_𝑛=𝑉_−𝑡 (22) and for 2≤𝑠≤𝑘 let 𝑥(𝑖+𝑠−1,𝑘+1−𝑠,𝑠−1)+𝑊_𝑛=𝑉_{3−𝑖−𝑠}. (23)
Setting 𝑊=𝑊_𝑛 and using (5) and (7) there are 𝑘+𝑖−1 vectors and 𝑚_𝑛 such that 𝑥(𝑖,𝑘,1)∈𝑈 satisfy, for 𝑢=1,2, 𝑇^𝑚𝑛_𝑢(𝑥(𝑖,𝑘,1))∈𝑉_𝑢. (24) And the remaining vectors chosen in this step satisfy the following displayed formulas. (Clearly when 𝑖=1 only (28) matters, whereas when 𝑘=1 only (29) matters.)
For 0≤𝑡≤𝑖−2 we have that 𝑥(𝑖−1−𝑡,𝑘+𝑡,𝑘+𝑖−1−𝑡)∈𝑉_−𝑡𝑇^𝑚𝑛_𝑢(𝑥(𝑖−1−𝑡,𝑘+𝑡,𝑘+𝑖−1−𝑡))∈𝑊_𝑛for0≤𝑡≤𝑖−2. (25)
For 2≤𝑠≤𝑘 we have 𝑥(𝑖+𝑠−1,𝑘+1−𝑠,𝑠)∈𝑉_{3−𝑖−𝑠} and 𝑇^𝑚𝑛_𝑢(𝑥(𝑖+𝑠−1,𝑘+1−𝑠,𝑠))∈𝑊_𝑛for2≤𝑠≤𝑘. (26)
We now choose a sufficiently small open set 𝑊_𝑛+1=𝐺_𝑝𝑛+1 with 𝑝_𝑛<𝑝_𝑛+1 and such that 𝑥−𝑥(𝑖,𝑘,1)∈𝑊_𝑛+1 implies that 𝑇^𝑚𝑛_𝑢(𝑥)∈𝑦_𝑛,𝑢+𝐺_𝑛, (27) and for the other vectors 𝑥(𝑐,𝑎,𝑏) with 𝑐≠𝑖 chosen at this stage we also have that 𝑥−𝑥(𝑐,𝑎,𝑏)∈𝑊_𝑛+1 implies that 𝑇^𝑚𝑛_𝑢(𝑥)∈𝑊_𝑛 (28) for 𝑢=1,2. Moreover, we also have that for all vectors 𝑥(𝑐,𝑎,𝑏) chosen at this stage 𝑥(𝑐,𝑎,𝑏)+𝑊_𝑛+1⊂𝑥𝑐,𝑎₋,𝑏₋+𝑊_𝑛, (29) where (𝑎₋,𝑏₋) is the immediate predecessor of (𝑎,𝑏) in the lexicographic order. When 𝑎=1=𝑏 we have that 𝑥(𝑖,1,1)+𝑊_𝑛+1⊂𝑓_𝑛+𝐺_𝑛. (30)
Since {𝑊_𝑛∶𝑛∈ℕ} is also a local basis of 0 and 𝑋 is complete, it follows from (29) and (30) that for each 𝑖 the sequence {𝑥(𝑖,𝑘,𝑗)} converges to a vector 𝑥_𝑖. Moreover, (27) and (28) and 𝑛=(𝑘+𝑖−1)(𝑘+𝑖)/2+1−𝑖 imply that 𝑇^𝑚𝑛_𝑢𝑥_𝑖−𝑦_𝑛,𝑢∈𝑊_𝑛,𝑇^𝑚𝑛_𝑢𝑥_𝑞∈𝑊_𝑛 (31) for 𝑞≠𝑖.
To finish the proof we have to prove that if 𝑥=∑^𝑀_𝑖=1𝜆_𝑖𝑥_𝑖 with some 𝜆_𝑖≠0, then 𝑥 is disjoint hypercyclic. To that end choose 𝜆_𝑖0 such that |𝜆_𝑖0|≥𝜆_𝑖 for 𝑖=1,…,𝑀. Since nonzero multiples of disjoint hypercyclic vectors are also disjoint hypercyclic, we may assume that 𝜆_𝑖0=1. For 𝑀<𝑘 and 𝑛=(𝑖₀+𝑘−1)(𝑖₀+𝑘)/2+1−𝑖₀ we have that 𝑥_𝑖0∈𝑥(𝑖₀,𝑘,1)+𝑊_𝑛+1 and for 𝑖≠𝑖₀ and 𝑖≤𝑀 we have that 𝑥_𝑖∈𝑥(𝑖,𝑎,𝑏)+𝑊_𝑛+1, where the vector 𝑥(𝑖,𝑎,𝑏) is obtained in the 𝑛 step. Therefore for 𝑢=1,2 we use (31) to get 𝑇^𝑚𝑛_𝑢𝑥_𝑖0+_{𝑖≠𝑖0,𝑖≤𝑀}𝜆_𝑖𝑥_𝑖−𝑦_𝑛,𝑢∈𝐺_𝑛+𝑊_𝑛+𝑊_𝑛+^𝑀−1⋯𝑊_𝑛 (32) since 𝜆_𝑖≤1 for 𝑖≠𝑖₀ and the open sets have been chosen to be balanced. Thus we have shown that 𝑥 is a disjoint hypercyclic vector.

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.

Corollary 9. Let 𝑋 be 𝑙^𝑝(𝕂) for 1≤𝑝<∞ or 𝑐₀(𝕂). If 𝑇₁,…,𝑇_𝑁 are disjoint hypercyclic weighted shifts, then they have a dense linear manifold of disjoint hypercyclic vectors.

Proof. Proposition 6 and Theorem 8 provide the proof.

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 𝑁-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 𝑇∈ℬ(𝑋) 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 𝑇∈ℬ(𝑋) is supercyclic if there is an 𝑥∈𝑋 such that {𝜆𝑇^𝑘𝑥∶𝑘∈ℕ,𝜆scalar} is dense in 𝑋. Chapter 9 of [8] treats supercyclicity. A supercyclic subspace for 𝑇∈ℬ(𝑋) 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].

A disjoint hypercyclic subspace for 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) is an infinite dimensional subspace whose nonzero vectors are disjoint hypercyclic. Proposition 3.7 of [18] 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 [23]; see also [22].

Question 1. Are the Disjoint Blow-up/Collapse property and its strong version equivalent?

Question 2. Given 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) that satisfy the Disjoint Blow-up/Collapse property, can we add some 𝑇_𝑁+1 such that the resultant 𝑁+1-tuple still satisfies the Disjoint Blow-up/Collapse property?

Question 3. Given 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) which are disjoint hypercyclic, can we add some 𝑇_𝑁+1 such that the resultant 𝑁+1-tuple is still disjoint hypercyclic?

Question 4. Assume that 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) have a disjoint hypercyclic space. Can we add some 𝑇_𝑁+1 such that the resultant 𝑁+1-tuple still has a disjoint hypercyclic subspace?
In the last three questions 𝑋 is either 𝑙^𝑝(𝕂) for 1≤𝑝<∞ or 𝑐₀(𝕂).

Question 5. Which weighted shifts 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) have a disjoint hypercyclic space?

Question 6. Which weighted shifts 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) are disjoint supercyclic?

Question 7. Which weighted shifts 𝑇₁,…,𝑇_𝑁∈ℬ(𝑋) 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].

Acknowledgment

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.