Existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing
aa r X i v : . [ m a t h . F A ] F e b EXISTENCE OF DISJOINT FREQUENTLY HYPERCYCLIC OPERATORS WHICHFAIL TO BE DISJOINT WEAKLY MIXING ¨OZG ¨UR MARTIN AND YUNIED PUIG
Abstract.
In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory,85 (2) (2016), 191-220] by showing the existence of disjoint frequently hypercyclic operators which fail tobe disjoint weakly mixing and, therefore, fail to satisfy the Disjoint Hypercyclicity Criterion. We also showthat given an operator T such that T ⊕ T is frequently hypercyclic, the set of operators S such that T, S are disjoint frequently hypercyclic but fail to satisfy the Disjoint Hypercyclicity Criterion is SOT dense inthe algebra of bounded linear operators. Introduction
Let N denote the set of non-negative integers, X be a separable and infinite dimensional Banach spaceover the real or complex scalar field K , and let B ( X ) denote the algebra of bounded linear operators on X .An operator T ∈ B ( X ) is called hypercyclic if there exists x ∈ X such that { T n x : n ∈ N } is dense in X andsuch a vector x is said to be a hypercyclic vector for T . By HC ( T ), we will denote the set of hypercyclicvectors for T .Disjointness in hypercyclicity is introduced independently by Bernal [2] and by B`es and Peris [7] in 2007.For N ≥
2, operators T , . . . , T N ∈ B ( X ) are called disjoint hypercyclic or d-hypercyclic if the direct sumoperator T ⊕ · · · ⊕ T N has a hypercyclic vector of the form ( x, . . . , x ) ∈ X N . Such a vector x ∈ X is calleda d-hypercyclic vector for T , . . . , T N . By d - HC ( T , . . . , T N ), we will denote the set of d-hypercyclic vectorsfor T , . . . , T N . If d - HC ( T , . . . , T N ) is dense, T , . . . , T N are called to be densely d-hypercyclic .It is well known that for T ∈ B ( X ), the set HC ( T ) is either empty or dense and it is non-empty ifand only if T is topologically transitive , that is for any two non-empty open U, V ⊂ X , there exists apositive integer n such that U ∩ T − n ( V ) = ∅ . Similarly, B`es and Peris [7] proved that operators T , . . . , T N are densely d-hypercyclic if and only if they are d-topologically transitive , that is for any non-empty open U, V , . . . , V N ⊂ X , there exists a positive integer n such that U ∩ T − n ( V ) ∩ . . . ∩ T − nN ( V N ) = ∅ . In [17],contrary to the single operator case, Sanders and Shkarin showed the existence of d-hypercyclic operatorswhich are not densely d-hypercyclic and, therefore, fail to be d-topologically transitive.We next remind a necessary condition for d-hypercyclicity, a natural extension of the HypercyclicityCriterion which has played a significant role in linear dynamics. Note that for N = 1, the following definitiongives the single operator version of the Hypercyclicity Criterion. Definition 1.1. [7] Let ( n k ) be a strictly increasing sequence of positive integers. We say that T , . . . , T N ∈B ( X ) satisfy the d-Hypercyclicity Criterion with respect to ( n k ) provided there exist dense subsets X , X , . . . , X N of X and mappings S m,k : X m → X with 1 ≤ m ≤ N, k ∈ N satisfying(1.1) T n k m −→ k →∞ X , S m,k −→ k →∞ X m , and( T n k m S i,k − δ i,m Id X m ) −→ k →∞ X m (1 ≤ i ≤ N ).In general, we say that T , . . . , T N satisfy the d-Hypercyclicity Criterion if there exists some sequence ( n k )for which (1.1) is satisfied. Date : August 7, 2020.2020
Mathematics Subject Classification.
Primary 47A16.
Key words and phrases.
Hypercyclic operator, Frequently hypercyclic operator, Disjoint hypercyclicity, Disjoint frequenthypercyclicity.
Theorem 1.2. [7, Theorem 2.7] T , . . . , T N ∈ B ( X ) satisfy the d-Hypercyclicity Criterion if and only if foreach r ∈ N , the direct sum operators L ri =1 T , . . . , L ri =1 T N are d-topologically transitive on X r . In [6], B`es and Peris showed that an operator T ∈ B ( X ) satisfies the Hypercyclicity Criterion if andonly if it is weakly mixing , that is T ⊕ T is topologically transitive. An older result of Furstenberg [12] alsoshows that T is weakly mixing if and only if for each r ∈ N , the direct sum operator L ri =1 T is topologicallytransitive. In a landmark result, De la Rosa and Read [11] constructed a Banach space that supports ahypercyclic operator which is not weakly mixing, and thus fails to satisfy the Hypercyclicity Criterion.In the disjointness case, the picture is again different. We say T , . . . , T N ∈ B ( X ) are d-weakly mixing if T ⊕ T , . . . , T N ⊕ T N are d-topologically transitive on X . In [17], Sanders and Shkarin also showed thatevery Banach space supports d-weakly mixing operators which fail to satisfy the d-Hypercyclicity Criterion.Then combining the above-mentioned results, we have the following implications: d-Hypercyclicity Criteron ⇒ d-weakly mixing ⇒ d-topologically transitive ⇒ d-hypercyclic and non of the reverse implications hold.In this short note, we are interested in the disjoint version of a strong recurrence property of hypercyclicoperators, called frequent hypercyclicity, which is introduced by Bayart and Grivaux [1]. An operator in T ∈ B ( X ) is called frequently hypercyclic if there exists some x ∈ X such that for every non-empty openset U ⊂ X the set { n : T n x ∈ U } ⊂ N has positive lower density. Such a vector x is called a frequentlyhypercyclic vector for the operator T . Recall that the lower density of a set A ⊂ N is defined bydens A := lim inf N →∞ card { n ≤ N : n ∈ A } N .
Frequent hypercyclicity cleary implies hypercyclicity. In fact, Grosse-Erdmann and Peris [13] showed thatfrequently hypercyclic operators are weakly mixing and, therefore, they satisfy the Hypercyclicity Criterion.We say T , . . . , T N ∈ B ( X ) with N ≥ d-frequently hypercyclic if there exists a vector x in X suchthat the vector ( x, . . . , x ) is a frequently hypercyclic vector for the direct sum operator T ⊕ · · · ⊕ T N on X N . Clearly, d-frequent hypercyclicity implies d-hypercyclicity, however it is not clear whether d-frequenthypercyclicity implies d-weakly mixing or even d-topological transitivity (densely d-hypercyclicity). In thenext section, we will answer in the negative the following questions posed in [14]: Question 1.3. If T , T ∈ B ( X ) are d-frequently hypercyclic, must they be d-weakly mixing? Must theysatisfy the d-Hypercyclicity Criterion?2. d-Frequently hypercyclic operators which fail to be d-weakly mixing For any
T, T , . . . , T N ∈ B ( X ), let F HC ( T ) and d - F HC ( T , . . . , T N ) denote the sets of frequentlyhypercyclic vectors of T and d-frequently hypercyclic vectors of T , . . . , T N , respectively. Note that, if( f , . . . , f N ) ∈ HC ( T ⊕ . . . ⊕ T ) then the vectors f , . . . , f N must be linearly independent. By modifyingthe results of Sanders and Shkarin in [17], we first show the existence of d-frequently hypercyclic operatorswhich fail to be d-weakly mixing. Lemma 2.1.
Let
T, L be in B ( X ) and L be invertible. If S := L − T L , then f ∈ d - F HC ( T, S ) if and onlyif ( f, Lf ) ∈ F HC ( T ⊕ T ) .Proof. If f ∈ d - F HC ( T, S ), then for any two non-empty, open sets
U, V ⊂ X we havedens { n ∈ N : T n f ∈ U, S n f ∈ L − ( V ) } > . This gives that dens { n ∈ N : T n f ∈ U, T n Lf ∈ V } > f, Lf ) ∈ F HC ( T ⊕ T ).Now if we assume ( f, Lf ) ∈ F HC ( T ⊕ T ) and U, V ⊂ X are non-empty and open, then by invertibilityof L , we can say that dens { n ∈ N : T n f ∈ U, T n Lf ∈ L ( V ) } > . But, this implies dens { n ∈ N : T n f ∈ U, L − T n Lf ∈ V } > f ∈ d - F HC ( T, S ). (cid:3) Lemma 2.2.
Let T be in B ( X ) and ( f, g ) ∈ F HC ( T ⊕ T ) . -FREQUENTLY HYPERCYCLIC OPERATORS WHICH FAIL TO BE D-WEAKLY MIXING 3 (1) For any r ∈ N , ( T r f, g ) ∈ F HC ( T ⊕ T ) . (2) For any non-zero c ∈ K , ( f, f + cg ) ∈ F HC ( T ⊕ T ) .Proof. To prove the first statement, let
U, V ⊂ X be any two non-empty open sets and r ∈ N . We have thatdens { n ∈ N : T n f ∈ T − r ( U ) , T n g ∈ V } > , which implies dens { n ∈ N : T n ( T r f ) ∈ U, T n g ∈ V } >
0, and ( T r f, g ) ∈ F HC ( T ⊕ T ).For the second statement, let c ∈ K with c = 0 and U, V ⊂ X be non-empty and open sets. Choose u ∈ U and v ∈ V and define x := v − u . Choose ε > B ( u , ε ) ⊂ U and B ( v , ε ) ⊂ V where B ( x, r ) denotes the ball centered at x ∈ X with radius r > { n ∈ N : T n f ∈ B ( u , ε/ , T n g ∈ c − B ( x , ε/ } > . This implies dens { n ∈ N : T n f ∈ B ( u , ε/ , T n f + cT n g ∈ B ( v , ε ) } >
0, and therefore,dens { n ∈ N : T n f ∈ U, T n ( f + cg ) ∈ V } > . Thus, ( f, f + cg ) ∈ F HC ( T ⊕ T ). (cid:3) Theorem 2.3.
Let X be a separable Banach space and T ∈ B ( X ) such that T ⊕ T is frequently hypercyclicon X × X . Then, there exists S ∈ B ( X ) such that T, S are densely d-frequently hypercyclic but they fail tobe d-weakly mixing.Proof.
Let ( f, g ) ∈ F HC ( T ⊕ T ) where f and g are linearly independent. By the Hahn-Banach Theorem,there exists a linear functional λ in the dual space X ∗ such that λ ( f ) = 1 and λ ( g ) = 0. Define L ∈ B ( X )by Lx = x + λ ( x ) g for x ∈ X . Then L is an invertible operator with the inverse L − x = x − λ ( x ) g . Now,define S := L − T L .First, we show that
T, S are densely d-frequently hypercyclic. To this end, let U ⊂ X be non-empty andopen and choose r ∈ N such that T r f ∈ U and λ ( T r f ) = 0.Since ( f, g ) ∈ F HC ( T ⊕ T ), we have that ( T r f, g ) ∈ F HC ( T ⊕ T ) by Lemma 2.2(1) and ( T r f, T r f + λ ( T r f ) g ) ∈ F HC ( T ⊕ T ) by Lemma 2.2(2). The last expression means ( T r f, LT r f ) ∈ F HC ( T ⊕ T ) whichin turn gives that T r f ∈ d - F HC ( T, S ) by Lemma 2.1.Now, in order to reach a contradiction, assume that T ⊕ T , S ⊕ S are d-hypercyclic. Then there exists a( x, y ) ∈ X × X such that the set { ( T n x, T n y, S n x, S n y ) : n ≥ } = { ( T n x, T n y, L − T n Lx, L − T n Ly ) : n ≥ } is dense in X . Therefore, ( x, y, Lx, Ly ) ∈ HC ( T ⊕ T ⊕ T ⊕ T ).As in Lemma 2.2, from the last statement we can derive( x, y, Lx − x, Ly − y ) ∈ HC ( T ⊕ T ⊕ T ⊕ T ) . This implies ( x, y, λ ( x ) g, λ ( y ) g ) ∈ HC ( T ⊕ T ⊕ T ⊕ T ), or in particular, ( λ ( x ) g, λ ( y ) g ) ∈ HC ( T ⊕ T ). But,this is a contradiction since the vectors λ ( x ) g and λ ( y ) g are linearly dependent. (cid:3) In [10], De la Rosa et al. showed that any complex separable infinite-dimensional Banach space with anunconditional Schauder decomposition supports an operator T such that T ⊕ T is frequently hypercyclic.Indeed, T has a perfectly spanning set of eigenvectors associated to unimodular eigenvalues, so does T ⊕ T .Hence, we have the following. Corollary 2.4.
Every complex separable infinite-dimensional Banach space with an unconditional Schauderdecomposition supports a densely d-frequently hypercyclic tuple of operators which fail to be d-weakly mixingand, therefore, fail to satisfy the d-Hypercyclicity Criterion.
Chan [8] showed that the hypercyclic operators on a separable infinite dimensional Hilbert space form adense subset of the algebra of continuous linear operators in the strong operator topology and, later, thisresult is extended to Frechet spaces by B`es and Chan [3]. In [15], the first author and Sanders showed thatfor any d-hypercyclic T , . . . , T N ∈ B ( X ) the set of operators S ∈ B ( X ) such that T , . . . , T N , S remain tobe d-hypercyclic is SOT-dense in X . Motivated by these results, we give the next theorem. ¨OZG¨UR MARTIN AND YUNIED PUIG Theorem 2.5.
Let X be a separable Banach space and T ∈ B ( X ) such that T ⊕ T is frequently hypercyclicon X × X . Then the set { S ∈ B ( X ) : T, S are d-frequently hypercyclic but do not satisfy the d-Hypercyclicity Criterion } is SOT-dense in B ( X ) .Proof. Let T ∈ B ( X ) so that T ⊕ T is frequently hypercyclic on X × X . Let A ∈ B ( X ) be arbitrary and U e ,...,e N ,ǫ := { B ∈ B ( X ) : k B ( e i ) − A ( e i ) k < ǫ, ≤ i ≤ N } be a SOT-neighborhood of A where e , . . . , e N ∈ X are linearly independent and ǫ > X supports a frequently hypercyclic operator, X is infinite dimensional and one can find f , . . . , f N ∈ X such that k f i − A ( e i ) k < ǫ for 1 ≤ i ≤ N , and e , . . . , e N , f , . . . , f N are linearly independent.Now choose x , . . . , x N , T x , . . . , T x N ∈ F HC ( T ) and ( f, g ) ∈ F HC ( T ⊕ T ) so that the set I := { e , . . . , e N , f , . . . , f N , x , . . . , x N , T x , . . . , T x N , f, g } is linearly independent.For each 1 ≤ i ≤ N , pick x ∗ i , y ∗ i ∈ X ∗ such that x ∗ i ( e i ) = x ∗ i ( x i ) = 1 with x ∗ i ≡ I\{ e i , x i } and y ∗ i ( f i ) = y ∗ i ( T x i ) = 1 with y ∗ i ≡ I\{ f i , T x i } . Let E := span { e , . . . , e N , f , . . . , f N } , F := span { x , . . . , x N , T x , . . . , T x N } , and Z := T Ni =1 (ker x ∗ i ∩ ker y ∗ i ). Then, X = E ⊕ Z = F ⊕ Z .Now, note that ( f, g ) ∈ F HC ( T ⊕ T ) ∩ ( Z × Z ). By the Hahn-Banach theorem, we can choose a λ ∈ X ∗ such that λ ( f ) = 1 and λ ( g ) = 0 and define L ∈ B ( X ) as follows:For any x ∈ X = E ⊕ Z , there exist unique y ∈ E and z ∈ Z such that x = y + z with y in the form y = N X j =1 α j e j + N X j =1 β j f j . Then define L ( x ) as L ( x ) = N X j =1 α j x j + N X j =1 β j T x j + z + λ ( z ) g. It is easy to see that L is invertible where for any x ∈ X = F ⊕ Z with x = y + z , y = P Nj =1 α j x j + P Nj =1 β j T x j ∈ F , and z ∈ Z , we have L − ( x ) = N X j =1 α j e j + N X j =1 β j f j + z − λ ( z ) g. Now, if we define S := L − T L , then S ∈ U e ,...,e N ,ǫ since Se i = f i for 1 ≤ i ≤ N . By Lemma 2.2, ( f, g ) ∈ F HC ( T ⊕ T ) implies that ( f, f + λ ( f ) g ) ∈ F HC ( T ⊕ T ). Since we also have that f ∈ Z , Lf = f + λ ( f ) g ,thus ( f, Lf ) ∈ F HC ( T ⊕ T ) and, as before, f ∈ d - F HC ( T, S ) by Lemma 2.1.Lastly, we need to show that
T, S do not satisfy the Disjoint Hypercyclicity Criterion. By Theorem 1.2, it isenough to show that L N +2 i =1 T, L N +2 i =1 S cannot be disjoint topologically transitive. By way of contradiction,assume ( u , . . . , u N +2 ) be a disjoint hypercyclic vector for the direct sums L N +2 i =1 T, L N +2 i =1 S . This meansthat { ( T n u , . . . , T n u N +2 , S n u , . . . , S n u N +2 ) : n ≥ } is dense in X N +4 . This, in turn, implies that( u , . . . , u N +2 , Lu , . . . , Lu N +2 ) ∈ HC N +4 M i =1 T ! . By Lemma 2.2, we conclude that( u , . . . , u N +2 , Lu − u , . . . , Lu N +2 − u N +2 ) ∈ HC N +4 M i =1 T ! , or(2.1) ( Lu − u , . . . , Lu N +2 − u N +2 ) ∈ HC N +2 M i =1 T ! . -FREQUENTLY HYPERCYCLIC OPERATORS WHICH FAIL TO BE D-WEAKLY MIXING 5 Now, it is enough to observe that, for 1 ≤ i ≤ N + 2, we have Lu i − u i ∈ span { e , . . . , e N , f , . . . , f N , x , . . . , x N , T x , . . . , T x N , g } , and, therefore, the set { Lu i − u i : 1 ≤ i ≤ N + 2 } is linearly dependent, contradicting (2.1). (cid:3) We remind the reader that determining whether T ⊕ T is frequently hypercyclic whenever T is frequentlyhypercyclic is still an open problem since Bayart and Grivaux [1] posed it for the first time in 2006. Therefore,we cannot remove the condition of frequent hypercyclicity of T ⊕ T in Theorem 2.3 and Theorem 2.5. However,we have a different panorama concerning U -frequent hypercyclicty and d-reiterative hypercyclicity. We getthese notions when we replace the positive lower density condition in the definition of frequent hypercyclictyby positive upper density and positive upper Banach density, respectively (see [5]). A recent result byErnst et al. [9], asserts that T is U -frequently hypercyclic (reiteratively hypercyclic) if and only if T ⊕ T is U -frequently hypercyclic (reiteratively hypercyclic). Now, observe that the proofs of Theorem 2.3 andTheorem 2.5 adapt easily to the notions of U -frequent hypercyclicity and reiterative hypercyclicity withoutany conditions on T ⊕ T . To the best of our knowledge, the following question is open: Question 2.6.
Does every separable Banach space support a reiteratively hypercyclic operator?In [4], B`es et al. showed the existence of a mixing operator T such that T, T are not d-mixing and askedwhether there exists a mixing operator T such that T, T are not even d-topologically transitive. Using aresult in ergodic Ramsey theory, the second author [16] answered this question in the positive by showingthat the same operator T given by B`es et al. is also chaotic, and T, T fail to be d-topologically transitive.So, we pose the following question: Question 2.7.
Is there an example of a frequently hypercyclic operator T for which T, T are not d-topologically transitive?We end this note by restating the following open question which was posed also in [14]: Question 2.8. If T , T ∈ B ( X ) are d-frequently hypercyclic, must they be densely d-hypercyclic (equiva-lently, d-topologically transitive)? References [1] F. Bayart and S. Grivaux, Frequently hypercyclic operators,
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