International Mathematical Forum, Vol. 7, 2012, no. 52, 2597 - 2602

On Syndetically Hypercyclic Tuples

Mezban Habibi

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

P. O. Box 181 40, Lidingo, Stockholm, Swedenhabibi.m@iaudehdasht.ac.ir

Abstract

In this paper we will give some conditions for a tuple of operatorsor 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 F be a topological vector space(TV S) and T1,T2,...,Tn are continuous map-ping on F , and T = (T1, T2, ..., Tn) be a tuple of operators T1,T2,...,Tn.Thetuple T is weakly mixing, if and only if, For any pair of non-empty open sub-sets U ,V in X, and for any syndetic sequences {mk,1}, {mk,2}, ..., {mk,n} withSupk(nk+1,j − nk,j) < ∞ for j = 1, 2, ..., n, then there exist mk,1, mk,2, ..., mk,n

such that Tmk,1

1 Tmk,2

2 ...Tmk,nn U ∩ V �= ∅, also, if and only if, it suffices in pre-

vious condition, to consider only those sequences mk,1, mk,2, ..., mk,n for whichthere is some m1 ≥ 1, m2 ≥ 1,...,mn ≥ 1 with mk,1 ∈ {mj , 2mj} for all k andall j. Reader can see [1 −−10] for some information.

2 Main Results

Let X be a metrizable and complete topological vector space(F−space) andT = (T1, T2, ..., Tn) is an n-tuple of operators, then we will let

F = {T1k1T2

k2...Tnkn : ki ≥ 0}

be the semigroup generated by T . For x ∈ X we take

Orb(T , x) = {Sx : S ∈ F} = {T1k1T2

k2 ...Tnkn(x) : ki ≥ 0, i = 1, 2, ..., n}.

2598 M. Habibi

The set Orb(T , x) is called, orbit of vector x under T and Tuple T = (T1, T2, ..., Tn)is called hypercyclic pair if the set Orb(T , x) is dense in X , that is

Orb(T , x) = {T1k1T2

k2 ...Tnkn(x) : ki ≥ 0, i = 1, 2, ..., n} = X .

The tuple T = (T1, T2, ..., Tn) is called topologically mixing if for any givenopen subsets U and V of X , there exist positive numbers K1, K2,...,Kn suchthat

T1k1,iT2

k2,i ...Tn,ikn,i(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 X ifthere exists some x ∈ X such that its orbit is dense in X, that is

Orb({Tn}n≥0, x) = Orb({x, T1x, T2x, ...} = X

In this case the vector x is called hypercyclic vector for the sequence {Tn}n≥0.Note that, if {Tn}n≥0 is a hypercyclic sequence of operators on X, then Xis necessarily separable. Also note that, the sequence {T n} is a hypercyclicsequence on X, if and only if, the operator T is hypercyclic operator on X.T is 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, ..., Tn) is an n-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,...,{mj,n}∞j=1 such that:1. T

mj,1

1 Tmj,2

2 ...Tmj,nn → 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 ...Tmj,nn Sjz → z,

then T is a hypercyclic n-tuple.

A strictly increasing sequence of positive integers {nk}k is said to be syndeticsequence, if Supk(nk+1 − nk) < ∞.A tuple T = (T1, T2, ..., Tn) on a space X is called syndetically hypercyclic iffor any syndetic sequences of positive integers

{mk,1}k, {mk,2}k, ..., {mk,n}k

the sequence

{T mk,1

1 Tmk,2

2 ...Tmk,nn }k

On syndetically hypercyclic Ttuples 2599

is hypercyclic, in other hand, there is x ∈ X such that {T nkx : k ≥ 0} is densin X, that is,

{T mk,1

1 Tmk,2

2 ...Tmk,nn (x)} = X.

Given continuous linear operators T1,T2,...,Tn that is T1, T2, ..., Tn ∈ L(X),defined on a separable F -space X, also suppose that T = (T1, T2, ..., Tn) bea tuple of operators T1,T2,...,Tn, Then T satisfies the Hypercyclicity Crite-rion if and only if for any strictly increasing sequences of positive integers{mk,1}k, {nk,2}k, ..., {mk,n}k such that

Supk(mk+1,j − mk,j) < ∞

for all j, then the sequence {T mk,1

1 Tmk,2

2 ...Tmk,nn }k is hypercyclic. Also, for each

hypercyclic vector x ∈ X of T , there exists two strictly increasing sequence{mk}k,{nk}k such that

Supk(mk+1,j − nk,j) < ∞

for all j, and {T mk,1

1 Tmk,2

2 ...Tmk,nn }k is somewhere dense, but not dense in X,

That is, the tuple T = (T1, T2, ..., Tn) and the sequence {T mk,1

1 Tmk,2

2 ...Tmk,nn }k

do not share the same hypercyclic vectors. Let F be a Frechet space andT1, T2, ..., Tn are bounded linear operators on F , and T = (T1, T2, ..., Tn) be atuple of operators T1, T2, ..., Tn. The space F is called topologically mixing iffor any given open sets U and V , there exist positive numbers M1, M2,...,Mn

such that

Tmi,1

1 Tmi,2

2 ...Tmi,nn (U) ∩ V �= φ , ∀mi,j ≥ Mj , i = 1, 2, ..., n

Notice that, If the tuple T satisfies the hypercyclic criterion for syndetic se-quences, then T is topologically mixing tuple on space F . Let V be a topolog-ical vector space(TV S) and T1, T2, ..., Tn are bounded linear operators on V ,and T = (T1, T2, ..., Tn) be a tuple of operators T1, T2, ..., Tn. The tuple T iscalled weakly mixing if

T × T × ... × T : X × X × ... × X → X × X × ... × X

is topologically transitive.

Theorem 2.2 Let X be a topological vector space(TV S) and T1, T2, ..., Tn arecontinuous mapping on X, and T = (T1, T2, ..., Tn) be a tuple of operatorsT1, T2, ..., Tn. Then the following 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}, ..., {mk,n}, there exist m′

k,1, m′k,2, ..., m

′k,n such that

Tm′

k,1

1 Tm′

k,2

2 ...Tm′

k,nn (U) ∩ (V ) �= ∅

2600 M. Habibi

(iii). It suffices in (ii) to consider only those sequences {mk,1}, {mk,2}, ..., {mk,n}for which there is some m1 ≥ 1, m2 ≥ 1,...,mn ≥ 1 with

mk,j ∈ {mj, 2mj}

for all k and for all j.

proof (i) → (ii). Given {mk,1}, {mk,2}, ..., {mk,n} and U ,V satisfying thehypothesis of condition (ii), take

mj = Supk{mk+1,j − mk,j}

for all j and the n−product map

n−tims︷ ︸︸ ︷T × T... × T :

n−tims︷ ︸︸ ︷X × X... × X →

n−tims︷ ︸︸ ︷X × X... × X

is transitive, Then there is mk′,1, mk′,2, ..., mk′,n in N such that

(Tmk′,11 T

mk′,22 ...T

mk′,nn (U))

⋂((T

mk′′,11 )−1(T

mk′′,22 )−1...(T

mk′′,nn )−1(V )) �= ∅

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

so(T

mk′,1+mk′′,11 T

mk′,2+mk′′,22 ...T

mk′,n+mk′′,nn (U))

⋂(V )) �= ∅

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

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

{mk,j : k ∈ N} ∩ {n + 1, n + 2, ..., n + mj} �= ∅

If for all j we select m′k,j ∈ {mk,j : k ∈ N} ∩ {n + 1, n + 2, ..., n + mj} then

we have Tm′

k,1

1 Tm′

k,2

2 ...Tm′

k,nn (U) ∩ (V ) �= ∅, by this the proof of (i) → (ii) is

completed.The case (ii) → (iii) is trivial.Case (iii) → (i). Suppose that U , V1, V2 are non-empty open subsets of X,then there 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 �= ∅

On syndetically hypercyclic Ttuples 2601

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+ηn

n U ⊂ V1

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,andT

mk,1

1 Tmk,2

2 ...Tmk,nn U ∩ V1 = ∅, ∀k ∈ N

Now we haveT

mk′,1+η1

1 Tmk′,2+η2

2 ...Tmk′,n+ηn

n 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+ηn

n U) ∩ (Tmk,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

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2602 M. Habibi

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[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.

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

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Received: May, 2012