JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 146517 10.1155/2013/146517 146517 Research Article The Strong Disjoint Blow-Up/Collapse Property Salas Héctor N. Fošner Ajda Department of Mathematics University of Puerto Rico Mayagüez PR 00681 USA upr.edu 2013 4 11 2013 2013 09 05 2013 14 08 2013 15 08 2013 2013 Copyright © 2013 Héctor N. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let X be a topological vector space, and let (X) be the algebra of continuous linear operators on X. The operators T1,,TN(X) are disjoint hypercyclic if there is xX such that the orbit {(T1n(x),,TNn(x)):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.

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 (X) be the algebra of continuous linear operators on X.

The operator T(X) is hypercyclic if there is xX such that Orb(T,x):={Tkx:k} is dense in X. This concept is closely related to the concept of transitivity from topological dynamics.

Definition A.

The operator T(X) is topologically transitive if for each pair U, V of nonempty open subsets of X there is n such that UT-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 .

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 T(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 (1)T-n(W)V0,T-n(V1)W.

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 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  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 N2. The operators T1,,TN(X) are disjoint hypercyclic if there is an xX such that the orbit {(T1n(x),,TNn(x)):n} is dense in X××X. The vector x is called a disjoint hypercyclic vector for T1,,TN.

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

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

Definition D. Let N2. The operators T1,,TN(X) are disjoint topologically transitive if whenever V0,V1,,VNX are nonempty open sets, then there exists n such that (2)T1-n(V1)TN-n(Vn)V0.

The second one is included in their Proposition  2.4.

Definition E. Let N2. The operators T1,,TN(X) satisfy the Disjoint Blow-up/Collapse property if for each open neighbourhood of zero W and nonempty open subsets V0,V1,,VNX there exists n such that (3)T1-n(W)TN-n(W)V0,T1-n(V1)TN-n(VN)W.

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 N2. Let (nk) be a strictly increasing sequence of positive integers. The operators T1,,TN(X) satisfy the Disjoint Hypercyclicity Criterion with respect to (nk) provided there exist dense subsets X0,X1,,XN of X and mappings Sl,k:XlX with 1lN and k satisfying (4)Tlnk0pointwiseonX0,Sl,k0pointwiseonXl,TlnkSi,k-δi,lIdXi0pointwiseonXiwith1iN.

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

Proposition 2.3 of  says that T1,,TN(X) are disjoint topologically transitive if and only if the set of disjoint hypercyclic vectors for T1,,TN is a dense Gδ of X.

Proposition 2.4 of  says that if T1,,TN(X) satisfy the Disjoint Blow-up/Collapse property, then they are disjoint topologically transitive. This is the disjoint version of Theorem 1.2 of , but the authors of  assume that X is a Banach space.

Theorem 2.7 of  says that T1,,TN(X) satisfy the Disjoint Hypercyclicity Criterion if and only if for each r the operators T1,,T1r,,TN,,TNr are disjoint topologically transitive in (Xr).

Theorem 2.7 of  also gives another equivalence.

When N=1, 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 2N. Bès et al. show in Theorems 2.1 and 2.2 of  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 . 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 T1,,TN(X) satisfy the Disjoint Hypercyclicity Criterion, then T1,,TN 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,,TN 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,,TN(X) are said to satisfy the Strong Disjoint Blow-up/Collapse property if for all m1 whenever nonempty open sets W,V1-m,,V0,V1,,VN are given with W a neighborhood of 0, then there exists n such that (5)T1-n(W)TN-n(W)Vk-mfor1km,(6)T1-n(V1)TN-n(VN)W.

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 , 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,,TN(X) satisfy the Strong Disjoint Blow-up/Collapse property. If m1 and the nonempty open sets W,U,V1,,VN and V1-m,,V0 are given with W a neighborhood of 0, then there exists n such that (5) holds and (7)T1-n(V1)TN-n(VN)U.

Proof.

Let xjVj for 1jN and xU. Let W be an open set containing 0 such that W+WW and xj+W+WVj for 1jN and x+W+WU.

Set V-m=x+W and Vi=Vi for 1-mi0 and Vj=xj+W for 1jN. Since T1,,TN(X) satisfy the Strong Disjoint Blow-up/Collapse property, we have that there exist n and ziVi for -mi0 and wW such that Tjn(zi)W and Tjn(w)Vj for -mi0 and 1jN. It remains to check that (8)T1-n(V1)TN-n(VN)U. Let z=z-m+wx+W+WU and Tjn(z)Vj+WVj for 1jN. Then zT1-n(V1)TN-n(VN)U.

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 T1,,TN(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 V1-m,,V0,V1,,VNX be nonempty open sets. Assume that X0,X1,,XN are the dense subsets of X and Sl,k:XlX with 1lN the mappings given by the Disjoint Hypercyclic Criterion. Let zjVjX0 for 1-mj0 and yiViXi for 1iN. We now choose another open neighbourhood of zero W with WW and such that 1iN and (9)W++WNW,yi+W++WNVi. By hypothesis there exists nk so that Tink(zj)W, for all 1iN and 1-mj0, and, for 1lN, (10)Sl,k(yl)W,TlnkSi,k(yi)-δl,iyiW.

Thus (5) is satisfied. It remains to verify that (6) is also satisfied. For that we choose w0=i=1NSi,k(yi) and it follows that (11)w0T1-nk(V1)TN-nk(VN)W.

The following proposition has an immediate proof, and it is often used when studying weighted shifts. Let 𝕂 be either or , and let X be lp(𝕂) for 1p< or c0(𝕂). Let {ei:i𝕂} be the canonical basis of X and for xX we denote by x,ei*=ei*(x), where ei*X* and ei*(ej)=δi,j. We say that the vector y  dominates the vector x if (12)|x,ei||y,ei|i.

Proposition 5.

Let T be a weighted shift. If the vector a dominates the vector b, then for all n, (13)Tn(b)Tn(a).

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 lp(𝕂) for 1p< or c0(𝕂). If T1,,TN 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 N=2 and for l2() or l2() which illustrate the general method. Let W={x:x<δ} and V2,V1,V0,,V-k for some k. Let {ei:i} and {ei:i} 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 {ei:iA} such that {x:x-xj<ϵ}V-j for j=0,,k and A is a finite interval of either or and ϵ<1/2. Let M=max{||x-j||+1:j=0,,k}. Let us choose z0 such that z,ei=M if iA and 0 otherwise. Set V^0={x:x-z<ϵ}. Apply the Disjoint Blow-up/Collapse property to W and V2, V1, and V^0. This means that there is an arbitrarily large n such that (14)V^0T1-n(W)T2-n(W),WT1-n(V1)T2-n(V2).

Case  1  (l2()). If n is large enough, we have that T1n(x)=0=T2n(x) for all x in the span of {ei:iA}, in particular for x0,,x-k. Thus (15)V-jT1-n(W)T2-n(W) for j=0,,k.

Case  2  (l2()). There exists zV^0 such that T1n(z)<δ and T2n(z)<δ. Since ϵ<1/2 and ei=1, we have for j=0,,k the following: (16)|z,ei||z0,ei|-(z-z0)|z,ei|M-12>x-j|x-j,ei| for iA; but for iA,|z,ei|0=|x-j,ei|. 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  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 T(X) is hypercylic, Herrero  and Bourdon , independently, showed that T has a dense linear manifold of hypercyclic vectors. (See also page 53 of .) If T1,,TN are disjoint hypercyclic, it is not known whether their set of disjoint hypercyclic vectors is dense in X, page 115 of . In view of the above results, the authors of  pose their Problem 3.6 which is the following.

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

In the following section we give a partial affirmative answer whenever T1,,TN 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,,TN(X) satisfy the Disjoint Blow-up/Collapse property, they have a dense set of disjoint hypercyclic vectors which is a Gδ, Proposition 2.3 of .

Theorem 8.

Let X be a topological vector space, over either or , whose topology has a countable basis and is complete. Let T1,,TN(X) satisfy the Strong Disjoint Blow-up/Collapse property. Then T1,,TN 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 1ji+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)=xi will exist. The linear manifold of disjoint hypercyclic vectors is the span of the xis.

Let {(yk,1,yk,2):k,u=1,2} be dense in X×X, and let {zp:p} be dense in X. Let A1A2A3= be a partition of such that each Ap is an infinite set and define fj=zp whenever jAp.

Let {Gn:n} be a local basis of 0 such that each Gn is a balanced open set and Gn+1¯Gn.

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)fi+Gi for all i, and, moreover, x(i,k,j)fi+Gi for all k and 1ji+k. We also have that their limit xifi+Gi. In this way we ensure that {xi:i} is dense in X since for jAp we have that limjxj=zp.

Step  1. Let U=f1+G1 and Vu=y1,u+G1 for u=1,2. Then by (7) we have x(1,1,1)U and m1 such that for u=1,2(17)Tum1(x(1,1,1))Vu.

Since T1, T2 are continuous, there exists W2=Gp2 such that 1=p1<p2 and x-x(1,1,1)W¯2 implies that Tum1(x)Vu for u=1,2. In addition, since {Gk} is a local basis, we can choose W2 sufficiently small that x(1,1,1)+W¯2f1+G1.

Step  2. Let U=f2+G2 and Vu=y2,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=1,2(18)Tum2(x(2,1,1))Vu,Tum2(x(1,1,2))W2.

Let W3=Gp3 be such that p2<p3 and x-x(2,1,1)W¯3 implies that Tum2(x)-y2,uW2, and if x-x(1,1,2)W¯3, then Tum2(x)W2. Also we can choose W3 such that x(1,1,2)+W¯3V0.

Step  n. Let (i,k) be such that n=(i+k-1)(i+k)/2+1-i.

The open set Wn=Gpn has been chosen in the previous step. Let (19)Vu=yn,u+Gnforu=1,2.

Case  (k=1).  We have that 1<i since 3n. Set U=fi+Gi and for 0ti-2 let (20)x(i-1-t,1+t,i-1-t)+Wn=V-t.Case  (i=1).  Set U=x(1,k-1,k)+Wn and for 2sk let (21)x(s,k+1-s,s-1)+Wn=V2-s.

Case  (1<k,1<i).  Set U=x(i,k-1,i+k-1)+Wn and for 0ti-2 let (22)x(i-1-t,k+t,k+i-2-t)+Wn=V-t and for 2sk let (23)x(i+s-1,k+1-s,s-1)+Wn=V3-i-s.

Setting W=Wn and using (5) and (7) there are k+i-1 vectors and mn such that x(i,k,1)U satisfy, for u=1,2, (24)Tumn(x(i,k,1))Vu. 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 0ti-2 we have that x(i-1-t,k+t,k+i-1-t)V-t(25)Tumn(x(i-1-t,k+t,k+i-1-t))Wn0000000000000g0000000for0ti-2.

For 2sk we have x(i+s-1,k+1-s,s)V3-i-s and (26)Tumn(x(i+s-1,k+1-s,s))Wnfor2sk.

We now choose a sufficiently small open set Wn+1=Gpn+1 with pn<pn+1 and such that x-x(i,k,1)W¯n+1 implies that (27)Tumn(x)yn,u+Gn, and for the other vectors x(c,a,b) with ci chosen at this stage we also have that x-x(c,a,b)W¯n+1 implies that (28)Tumn(x)Wn for u=1,2. Moreover, we also have that for all vectors x(c,a,b) chosen at this stage (29)x(c,a,b)+W¯n+1x(c,a-,b-)+Wn, where (a-,b-) is the immediate predecessor of (a,b) in the lexicographic order. When a=1=b we have that (30)x(i,1,1)+W¯n+1fn+Gn.

Since {Wn: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 xi. Moreover, (27) and (28) and n=(k+i-1)(k+i)/2+1-i imply that (31)Tumn(xi)-yn,uWn,Tumn(xq)Wn for qi.

To finish the proof we have to prove that if x=i=1Mλixi with some λi0, then x is disjoint hypercyclic. To that end choose λi0 such that |λi0|λi for i=1,,M. Since nonzero multiples of disjoint hypercyclic vectors are also disjoint hypercyclic, we may assume that λi0=1. For M<k and n=(i0+k-1)(i0+k)/2+1-i0 we have that xi0x(i0,k,1)+W¯n+1 and for ii0 and iM we have that xix(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 (32)Tumn(xi0+ii0,iMλixi)-yn,uGn+Wn+Wn+WnM-1 since λi1 for ii0 and the open sets have been chosen to be balanced. Thus we have shown that x is a disjoint hypercyclic vector.

The authors of  comment that they do not know the answer to Problem 3.6 of  (Problem  G) even when the operators are weighted shifts.

Corollary 9.

Let X be lp(𝕂) for 1p< or c0(𝕂). If T1,,TN 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  which also characterize disjoint hypercyclicity in terms of their weight sequence.

First we recall the relevant definitions. A hypercyclic subspace for T(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 . Chapter 8 of  and Chapter 10 of  give the fundamentals and history of hypercyclic subspaces.

The operator T(X) is supercyclic if there is an xX such that {λTkx:k,λscalar} is dense in X. Chapter 9 of  treats supercyclicity. A supercyclic subspace for T(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 .

A disjoint hypercyclic subspace for T1,,TN(X) 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 . 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 T1,,TN(X) that satisfy the Disjoint Blow-up/Collapse property, can we add some TN+1 such that the resultant N+1-tuple still satisfies the Disjoint Blow-up/Collapse property?

Question 3.

Given T1,,TN(X) which are disjoint hypercyclic, can we add some TN+1 such that the resultant N+1-tuple is still disjoint hypercyclic?

Question 4.

Assume that T1,,TN(X) have a disjoint hypercyclic space. Can we add some TN+1 such that the resultant N+1-tuple still has a disjoint hypercyclic subspace?

In the last three questions X is either lp(𝕂) for 1p< or c0(𝕂).

Question 5.

Which weighted shifts T1,,TN(X) have a disjoint hypercyclic space?

Question 6.

Which weighted shifts T1,,TN(X) are disjoint supercyclic?

Question 7.

Which weighted shifts T1,,TN(X) have a disjoint supercyclic space?

More unsolved questions on disjoint hypercyclicity can be found in [1012, 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.

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