Pure Mathematical Sciences, Vol. 3, 2014, no. 1, 17 - 21HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/pms.2014.359

On Hypercyclicity ∞-Tuples

of Commutative Bounded Linear Operators

S. Nasrin Hoseini M.

Department of MathematicsGenaveh Branch, Islamic Azad University, Genaveh, Iran

P. O. Box 7561738455, Borazjan, Iran

Mezban Habibi

Department of MathematicsMinistry of Education, Fars province organization, Shiraz, IranP. O. Box 181 56, Svalsvagen 1, Lidingo, Stockholm, Sweden

Copyright c© 2014 S. Nasrin Hoseini M. and Mezban Habibi. This is an open accessarticle distributed under the Creative Commons Attribution License, which permits unre-stricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

AbstractIn this paper we study the epsilon hypercyclicity on ∞-tuple makes

with commutative bounded linear operators on Banach spaces.

Mathematics Subject Classification: 47A16, 37A25

Keywords: Hypercyclic vector, ∞-tuple, Hypercyclicity Criterion, Ba-nach space, Epsilon hypercyclicity

1 Introduction

If T1, T2, ..., Tn, ... be commutative bounded linear mappings on a Banach spaceX and T = (T1, T2, ..., Tn, ...), by T (x) or Tm1

1 Tm22 ...Tmn

n ...(x) we meanSupn→∞{T m1

1 Tm22 ...Tmn

n (x) : m1, m2, ..., mn ≥ 0}So

T (x) = Tm11 Tm2

2 ...Tmnn ...(x)

= Supn→∞{T m11 Tm2

2 ...Tmnn (x) : m1, m2, ..., mn ≥ 0}

18 S. Nasrin Hoseini M. and Mezban Habibi

Definition 1.1 Let T1, T2, ..., Tn, ... be commutative bounded linear opera-tors on a Banach space X . For ∞-tuple T = (T1, T2, ..., Tn, ...), put

Γ = {T m11 Tm2

2 ...Tmnn ... : m1, m2, ..., mn, ... ≥ 0}

For x ∈ X , the orbit of x under T is the set Orb(T , x) = {S(x) : S ∈ Γ}, thatis

Orb(T , x) = {T m11 Tm2

2 ...Tmnn ...(x) : m1, m2, ..., mn... ≥ 0}

The vector x is called hypercyclic vector for T and ∞-tuple T is called hyper-cyclic ∞-tuple, if the set Orb(T , x) is dense in X , that is

Orb(T , x) = {T m11 Tm2

2 ...Tmnn ...(x) : m1, m2, ..., mn... ≥ 0} = X

Also if ε be a number in (0, 1) and x be a vector of X . The vector x is calledε-hypercyclic vector for ∞-tuple T = (T1, T2, ..., Tn, ...) and the ∞-tuple T iscalled ε-hypercyclic ∞-tuple, if for every non zero vector y ∈ X , there existsome integers m1, m2, ..., mn, ... such that

‖ Tm11 Tm2

2 ...Tmn2 ...x − y ‖< ε ‖ y ‖

All operators in this paper are commutative operators on a Banach space X .Readers can see [1 −−10] for more information.

2 Main Results

The bellow theorem namely the Hypercyclicity Criterion is useful theorem inoperator theory, that it used in papular theorem’s proof, if ∞-tuple T satisfythis theorem then we say that it satisfying the The Hypercyclicity Criterion.

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

mj,1

1 Tmj,2

2 ...Tmj,nn ... → 0 on Y as j → ∞,

2. There exist function {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 ∞-tuple.

Theorem 2.2 Let X be a separable Hilbert space and T = (T1, T2, ..., Tn, ...)be an ∞-tuple of commutative bounded linear operators on a Hilbert space X .If for every ε > 0, the ∞-tuple T is ε-hypercyclic, then T is a hypercyclic .

On hypercyclicity ∞-tuples of commutative bounded linear operators 19

Proof . Note that, if T is a Hypercyclic ∞-tuple, then σp(T∗) = φ, (T ∗ =

(T ∗1 , T ∗

2 , ..., T ∗n , ...)), also all spaces that admitted some hypercyclic operator,

are infinite dimensional spaces, so we can assume that X be infinite dimensionalspace. Suppose that U and V are subset of X . Give u ∈ U , v ∈ V twononzero element and δ > 0 so large that B(u, δ) ⊂ U and B(v, δ) ⊂ V so thatδ < Max{‖ u ‖, ‖ v ‖}. Take x such that x be an ε-hypercyclic for T withproperty ε < δ

6Max{‖u‖,‖v‖} , then there exist m1,0, m2,0, ..., mn,0, ... such that

‖ Tm1,0

1 Tm2,0

2 ...Tmn,0n ...x − u ‖< ε ‖ u ‖< δ

thus we haveTm1

1 Tm22 ...Tmn

n ...x ∈ USuppose on the contrary that there are only finitely many such integers

m1,1, m2,1, . . . , mn,1

m1,2, m2,2, . . . , mn,2

. . .

m1,t, m2,t, . . . , mn,t

. . .

As above, for each u′ ∈ X with

‖ u′ − u ‖< 2δ

3

there exist integersm1(u

′), m2(u′), ..., mn(u′), ...

which satisfies

‖ Tm1(u′)1 T

m2(u′)2 ...Tmn(u′)

n ...x − u′ ‖≤ ε ‖ u′ ‖ 2ε ‖ u ‖< δ

3

Since

‖ Tm1(u′)1 T

m2(u′)2 ...Tmn(u′)

n ...x−u′ ‖≤‖ Tm1(u′)1 T

m2(u′)2 ...Tmn(u′)

n ...x−u ‖ + ‖ u−u′ ‖< δ

we havemk(u

′) ∈ {m1,k, m2,k, ..., mn,k, ...}for k = 1, 2, ..., t, ... and the ball B(u, 2δ

3) is covered by a finite number balls

B(Tm1,1

1 Tm2,1

2 ...Tmn,1n x,

δ

3, ...)

20 S. Nasrin Hoseini M. and Mezban Habibi

B(Tm1,2

1 Tm2,2

2 ...Tmn,2n x,

δ

3, ...)

. . .

B(Tm1,t

1 Tm2,t

2 ...Tmn,tn x,

δ

3, ...)

. . .

Thus in an infinite dimensional space this is impossible. So there are infinitelymany integers as m1, m2, . . . , mn, ... with

‖ Tm11 Tm2

2 ...Tmnn ...x − u ‖< δ

Then there exist mi,k > mi,0 for k = 1, 2, ..., t, ... and i = 1, 2, ..., n, ... such that

Tm11 Tm2

2 ...Tmnn ...x ∈ V

ThusT

m1,1−m1,0

1 Tm2,2−m2,0

2 ...Tmn,n−mn,0n ...T

m1,0

1 Tm2,0

2 ...Tmn,0n ...x

is belong toV

⋂T

m1,1−m1,0

1 Tm2,2−m2,0

2 ...Tmn,n−mn,0n ...(U)

that is

Tm1,1

1 Tm2,2

2 ...Tmn,nn ...x ∈ (V

⋂T

m1,1−m1,0

1 Tm2,2−m2,0

2 ...Tmn,n−mn,0n ...)

Here it can be concluded that T is hypercyclic ∞-tuple.

References

[1] J. Bes, Three problem on hypercyclic operators, PhD thesis, Kent StateUniversity, 1998.

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

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

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

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

[6] 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.

On hypercyclicity ∞-tuples of commutative bounded linear operators 21

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

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

Received: May 1, 2013