Int. Journal of Math. Analysis, Vol. 8, 2014, no. 20, 951 - 955HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2014.4114

On Syndetically Hypercyclic ∞-Tuples

Mezban Habibi

Department of MathematicsDehdasht Branch, Islamic Azad University, Dehdasht, Iran

P.O. Box 181 40, Lidingo, Stockholm, Sweden

Copyright c© 2014 Mezban Habibi. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we will give some conditions for an ∞-tuple of operatorsor a tuple of weighted shifts to be Syndetically Hypercyclic.

Mathematics Subject Classification:.47A16, 47B37.

Keywords: Syndetically tuple, Hypercyclic tuple, Hypercyclic vector, Hy-percyclicity Criterion, Syndetically Hypercyclic

1 Introduction

Let B be a Banach space and T1,T2,... are commutative bounded linear mappingon B, the ∞-tuple T is an infinity component (T1, T2, ...), for each x ∈ Bdefined T (x) = T1

k1,jT2k2.j ...(x) = Supn{T1

k1T2k2...Tn

kn(x) : n ∈ N, kj ≥0, j = 1, 2, 3, ..., n}. Let πT = {T1

k1T2k2... : ki ≥ 0, i = 1, 2, 3, ...} be the

semigroup generated by T , for x ∈ B take Orb(T , x) = {Sx : S ∈ πT } ={T1

k1,jT2k2.j ...(x) : ki,j ≥ 0, i = 1, 2, ...}. The set Orb(T , x) is called, orbit of

vector x under T and ∞-Tuple T = (T1, T2, ...) is called hypercyclic ∞-tuple ifthe set Orb(T , x) is dense in B. Strictly increasing sequence of positive integers{mk}∞k=1 is said to be syndetic sequence, if Supk(nk+1−nk) < ∞. An ∞-tupleT is called weakly mixing if for any pair of non-empty open subsets U ,V ofB and any syndetic sequences {mk,1},{mk,2},... with Supk(mk+1,j − mk,j) <∞, j = 1, 2, ..., there exist mk,1, mk,2, ... such that T

mk,1

1 Tmk,2

2 ...(U)∩V �= ∅. An∞-tuple T = (T1, T2, ...) is called topologically mixing if for any given opensubsets U and V subsets of B, there exist positive numbers Ki, i = 1, 2, 3, ...,

952 M. Habibi

such that T1k1T2

k2T3k3...(U) ∩ V �= φ, ∀kj,i ≥ Ki, ∀j = 1, 2, ..., n. A sequence

of operators {Tn}n≥0 is said to be a hypercyclic sequence on B if there existssome x ∈ X such that its orbit under this sequence is dense in B, that isOrb({Tn}n≥0, x) = Orb({x, T1x, T2x, ...} = B, in this case the vector x is calledhypercyclic vector for the sequence {Tn}n≥0. Reader can see [1−−10] for moreinformation.

2 Main Results

An ∞-tuple T = (T1, T2, ...) on a space B is called syndetically hypercyclicif for any syndetic sequences of positive integers {mk,1}k, {mk,2}k, ..., the se-quence {T mk,1

1 Tmk,2

2 ...}k is hypercyclic, in other hand, there is x ∈ B such that

{T nkx : k ≥ 0} is dens in B, that is, {T mk,1

1 Tmk,2

2 ...(x)} = B. Let commutativecontinuous linear operators T1,T2,..., defined on a separable F -space B, Then∞-tuple T = (T1, T2, ...) satisfies the Hypercyclicity Criterion if and only if forany strictly increasing sequences of positive integers {mk,1}k, {nk,2}k, ... withproperties Supk(mk+1,j −mk,j) < ∞ for all j, then the sequence {T mk,1

1 Tmk,2

2 ...is hypercyclic. Notice that, If the tuple T satisfies the hypercyclic criterionfor syndetic sequences, then T is topologically mixing tuple on space B. Tis said to satisfy the Hypercyclicity Criterion if it satisfies the hypothesis ofbelow theorem.

Theorem 2.1 (The Hypercyclicity Criterion) Let X be a separable Ba-nach space and T = (T1, T2, ...) is an ∞-tuple of continuous linear mappingson X. If there exist two dense subsets Y and Z in X, and strictly increasingsequences {mj,1}∞j=1, {mj,2}∞j=1,... such that:1. T

mj,1

1 Tmj,2

2 ... → 0 on Y as j → ∞,2. There exist functions {Sj : Z → X} such that for every z ∈ Z, Sjz → 0,and T

mj,1

1 Tmj,2

2 ...Sj(z) → z,then T is a hypercyclic n-tuple.

Theorem 2.2 Let X be a topological vector space and T1, T2, ... are commuta-tive bounded linear mapping on X , and T = (T1, T2, ...) be an ∞-tuple of thoseoperators. The following conditions are equivalent:(i). T is weakly mixing.(ii). For any pair of non-empty open subsets U ,V in X , and for any syndeticsequences {mk,1}, {mk,2}, ..., there exist m′

k,1, m′k,2, ... such that

Tm′

k,1

1 Tm′

k,2

2 ...(U) ∩ (V ) �= ∅

(iii). It suffices in (ii) to consider only those sequences {mk,1}, {mk,2}, ... for

On syndetically hypercyclic ∞-tuples 953

which there is some m1 ≥ 1, m2 ≥ 1,... with mk,j ∈ {mj , 2mj} for all k andfor all j.

proof (i) → (ii). Given {mk,1}, {mk,2}, ... and U ,V satisfying the hypoth-esis of condition (ii), take mj = Supk{mk+1,j − mk,j} for all j and since∞−product map

n−tims︷ ︸︸ ︷T × T... × T :

n−tims︷ ︸︸ ︷X × X... × X →

n−tims︷ ︸︸ ︷X × X... × X

as n → ∞, is transitive, then there is mk′,1, mk′,2, ... in N such that

(Tmk′,11 T

mk′,22 ...(U))

⋂((T

mk′′,11 )−1(T

mk′′,22 )−1...(V )) �= ∅

∀mk′′,1 = 1, 2, ...,∀mk′′,2 = 1, 2, ...,∀mk′′,n = 1, 2, ...

so(T

mk′,1+mk′′,11 T

mk′,2+mk′′,22 ...(U))

⋂(V )) �= ∅

∀mk′′,1 = 1, 2, ...,∀mk′′,2 = 1, 2, ..., m, ..., ∀mk′′,n = 1, 2, ...

By the assumption on {mk,1}, {mk,2}, ...,for all j, we have

{mk,j : k ∈ N} ∩ {n + m1, n + m2, ..., n + mj , ...} �= ∅

If for all j we select m′k,j ∈ {mk,j : k ∈ N} ∩ {n + m1, n + m2, ..., n + mj , ...}

then we have Tm′

k,1

1 Tm′

k,2

2 ...(U) ∩ (V) �= ∅, by this the proof of (i) → (ii) iscompleted.The case (ii) → (iii) is trivial.Case (iii) → (i). Suppose that U , V1, V2 are non-empty open subsets of §, thenthere are {mk,1}, {mk,2}, ..., {mk,n, ...} in N such that

Tmk,1

1 Tmk,2

2 ...Tmk,nn ...U ∩ V1 �= ∅

Tmk,1

1 Tmk,2

2 ...Tmk,nn ...U ∩ V2 �= ∅.

This will imply that T is weakly mixing. Since (iii) is satisfied, then we cantake {mk,1}, {mk,2}, ..., {mk,n}, ... in N such that

Tmk,1

1 Tmk,2

2 ...Tmk,nn ...V1 ∩ V2 �= ∅

By continuity, we can find V1 ⊂ V1 open and non-empty such that

Tmk,1

1 Tmk,2

2 ...Tmk,nn ...V1 ⊂ V2.

Also there exist some mk′,1, mk′,2, ..., mk′,n, ... in N such that

Tmk′,1+η1

1 Tmk′,2+η2

2 ...Tmk′,n+ηnn ...U ⊂ V1

954 M. Habibi

for ηj = 0, mj we take mk,j = mk′,j + ηj, for all j, indeed we find strictlyincreasing sequences of positive integers mk,1, mk,2, ..., mk,n, ... such that mk,j ∈{mj , 2mj} for all j,and

Tmk,1

1 Tmk,2

2 ...Tmk,nn ...U ∩ V1 = ∅, ∀k ∈ N

Now we haveT

mk′,1+η1

1 Tmk′,2+η2

2 ...Tmk′,n+ηnn ...U

⋂V1 �= ∅

So the set

∅ �= Tmk,1

1 Tmk,2

2 ...Tmk,nn ...(T

m′k,1

1 Tm′

k,2

2 ...Tm′

k,nn ...U ∩ V1)

is a subset of

(Tmk′,1+η1

1 Tmk′,2+η2

2 ...Tmk′,n+ηnn ...U) ∩ (T

mk,1

1 Tmk,2

2 ...Tmk,nn ...V1)

then we haveT

mk,1

1 Tmk,2

2 ...Tmk,nn ...U

⋂V1 �= φ

and similarlyT

mk,1

1 Tmk,2

2 ...Tmk,nn ...U

⋂V2 �= φ

now, this is the end of proof.

References

[1] J. Bes and A. Peris, Hereditarily hypercylic operators, Jour. Func. Anal., 1 (167) (1999), 94-112.

[2] M. Habibi, n-Tuples and chaoticity, Int. Journal of Math. Analysis, 6 (14)(2012), 651-657 .

[3] M. Habibi, ∞-Tuples of Bounded Linear Operators on Banach Space, Int.Math. Forum, 7 (18) (2012), 861-866.

[4] M. Habibi and F. Safari, n-Tuples and Epsilon Hypercyclicity, Far EastJour. of Math. Sci. , 47 (2) (2010), 219-223.

[5] M. Habibi and B. Yousefi, Conditions for a tuple of operators to be topo-logically mixing, Int. Jour. of App. Math. , 23(6) (2010), 973-976.

[6] A. Peris and L. Saldivia, Syndentically hypercyclic operators, IntegralEquations Operator Theory, 51, No. 2 (2005) 275-281.

[7] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. , 347(1995), 993-1004.

On syndetically hypercyclic ∞-tuples 955

[8] B. Yousefi and M. Habibi, Syndetically Hypercyclic Pairs, Int. Math. Fo-rum , 5 (66) (2010), 3267 - 3272.

[9] B. Yousefi and M. Habibi, Hereditarily Hypercyclic Pairs, Int. Jour. ofApp. Math., 24(2) (2011)), 245-249.

[10] B. Yousefi and M. Habibi, Hypercyclicity Criterion for a Pair of WeightedComposition Operators, Int. Jour. of Appl. Math. , 24 (2) (2011), 215-219.

Received: January 3, 2014