paperno15-hoseinihabibisafarighezelbash-ijma

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 13, 643 - 650

Semi-Periodic Vector for n-Tuples of Operators

S. Nasrin Hoseini M.

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

P. O. Box 7561738455, Borazjan, Irannasrin hosseini [email protected]

Mezban Habibi

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

P. O. Box 181 40, Lidingo, Stockholm, [email protected]

Fatemeh Safari

department of mathematicsDehdasht Branch, Islamic Azad University, Dehdasht, Iran

P.O. Box 7571763111, Mamasani, [email protected]

Fatemeh Ghezelbash

Department of MathematicsMeymeh Branch, Islamic Azad University, Meymeh, Iran

P. O. Box 8196983443,Isfahan , [email protected]

Parvin Karami

department of mathematicsMamasani Branch, Islamic Azad University, Mamasani, Iran

P.O. Box 7571758379, Mamasani, [email protected]

Abstract

In this paper, we introduce fix points for a tuple of operators on a

644 S. N. Hoseini, M. Habibi, F. Safari, F. Ghezelbash and P. Karami

Banach space and give some Conditions for a vector to be a Fix pointfor the Tuple.

Mathematics Subject Classification: 37A25, 47B37

Keywords: Hypercylicity criterion, n-tuple, Fix point, Semi-Periodic vec-tor, Dense generalized kernel

1 Introduction

Let X be a Banach space and T1, T2, ..., Tn are commutative bounded linearoperators on X . By an n-tuple we mean the n-component T = (T1, T2, ..., Tn)so that ”n” is a finite positive integer number. For the tuple T = (T1, T2, ..., Tn)the set

F = {T1k1T2

k2...Tnkn : ki ≥ 0, i = 1, 2, ..., n}

is the semigroup generated by T . For x ∈ X take

Orb(T , x) = {Sx : S ∈ F}.

In other hand

Orb(T , x) = {T1k1T2

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

2 Preliminary Notes

Definition 2.1 The set Orb(T , x) is called, orbit of vector x under T andTuple T = (T1, T2, ..., Tn) is called hypercyclic tuple, if there is a vector x ∈ Xsuch that, 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 .

In this case, the vector x is called a hypercyclic vector for the tuple T .

Definition 2.2 Let X is a metric space with metric d. Take

η(X ) = P (X ) − {φ}

andCl(X ) = {x : x ⊂ X , x = closed, x �= φ}

Semi-periodic vector for n-tuples of operators 645

Consider an n-tuple T = (T1, T2, ..., Tn). An element x ∈ X is called a fixedpoint for T if there exist non-negative integer numbers m1, m2, ..., mn such that

Tm11 Tm2

2 ...Tmnn (x) = x

Fixed point for a functional is defining similarly.

Definition 2.3 Let X is a Banach space and x ∈ X, the vector x is calleda semi-periodic vector for tuple T = (T1, T2, ..., Tn) if the sequence

{T m11 Tm2

2 ...Tmnn (x)}

be semi-compact. In this case T is called semi-periodic tuple.

Definition 2.4 Let X is a Banach space and T = (T1, T2, ..., Tn) is a Tuple,then the dense generalized kernel of T is the set

⋃

k>0

(Ker(Tm1,k

1 Tm2,k

2 ...Tmn,kn )).

Note 2.5 All of operators in this paper are commutative bounded linearoperators on a Banach space. Also, note that by {j, i} we mean a number, thatwas showed by this mark and related with this indexes, not a pair of numbers.

Readers can see [1-9] for some information.

3 Main Results

Theorem 3.1 (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, on Z as j → ∞,

then T is a hypercyclic n-tuple.


Theorem 3.2 Let X be a separable Banach space and T = (T1, T2, ..., Tn)is an hypercyclic n-tuple of commutative continuous linear mappings on X .the tuple T satisfying the hypothesis of The Hypercyclicity Criterion, if thereis a subset S of X such that, all elements of S are semi-periodic vectors fortuple T .

proof. Since T is hypercyclic tuple, then we can choice a vector x ∈ Xsuch that Orb(T , x) is dense in X . Take

Uk = (0,1

k)

Vk = {x + u : u ∈ Uk}.Since the set of hypercyclic vectors for any tuple is dense in X , so let u1 ∈ U1

and the natural numbers m1,1, m2,1, ..., mn,1 with property

Tm1,1

1 Tm2,1

2 ...Tmn,1n (u1) ∈ V1.

Now, take u2 ∈ U2. Since

{T m1,1+11 T

m2,1+12 ...T

mn,1+1n (u2), T

m1,1+21 T

m2,1+22 ...T

mn,1+2n (u2), ...} = X

so there are m1,2, m2,2, ..., mn,2 such that

Tm1,2

1 Tm2,2

2 ...Tmn,2n (u2) ∈ V2.

Similarly, there are m1,t, m2,t, ..., mn,t such that

Tm1,t

1 Tm2,t

2 ...Tmn,tn (ut) ∈ Vk.

There are sequence {ut}∞k=1 of hypercyclic vectors and subsequences {m1,t}, {m2,t}, ..., {mn,t}of natural numbers, such that

limk→∞(uk) = 0

andT

m1,t

1 Tm2,t

2 ...Tmn,tn (ut) ∈ Vt.

Now we try to find subsequence

{m′1,t}, {m′

2,t}, ..., {m′n,t}

of{m1,t}, {m2,t}, ..., {mn,t}


such thatT

m′1,t

1 Tm′

2,t

2 ...Tm′

n,tn (u2)(Vt)

⋂(Ut) �= φ.

Suppose that V = Vk and U = Uk for any given k. If S be the set of allsemi-periodic vectors of T then S = X so

S⋂

(x + B(0,1

2k)) �= φ

in other word, we can take

ω ∈ S⋂

(x + B(0,1

2k))

indeed the orbit of{T m1,t

1 Tm2,t

2 ...Tmn,tn (ut)}

is semi-compact, so there subsequence

{η1,t}, {η2,t}, ..., {ηn,t}

of{m1,j}, {m2,j}, ..., {mn,j}

such that{T η1,t

1 Tη2,t

2 ...T ηn,tn (ω)}

is a convergence sequence. Suppose ω0 ∈ X and

Tη1,t

1 Tη2,t

2 ...T ηn,tn (ω) → ω0

as t → ∞. With replace ε = 12k

we have k0,

‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω) − ω0‖ ≤ 1

2k0

as ηj,t > k for j = 1, 2, ..., n and t = 1, 2, .... since x be a hypercyclic vector forS then there are natural numbers

η1,t, η2,t, ..., ηn,t

such that,

‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω) + ω‖ ≤ 1

2k.

Since lim(ui) = 0 as i → 0, then there is α such that

‖ uα ‖≤ 1

2k0 ‖ Tη1,t

1 Tη2,t

2 ...Tηn,tn ‖ .


For k, take ηi,t with property ηi,t > k and ηi,t > k0 for i = 1, 2, ..., n andt = 1, 2, ..., now we have

‖T η1,k

1 Tη2,k

2 ...Tηn,kn T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui) + ω‖

= ‖T η1,k

1 Tη2,k

2 ...Tηn,kn T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui)

+Tη1,k

1 Tη2,k

2 ...Tηn,kn (x)−T

η1,k

1 Tη2,k

2 ...Tηn,kn (x)+ω‖

= ‖T η1,k

1 Tη2,k

2 ...Tηn,kn (T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui) − x)+

Tη1,k

1 Tη2,k

2 ...Tηn,kn (x) + ω‖

≤ ‖T η1,k

1 Tη2,k

2 ...Tηn,kn (T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui) − x)‖+

‖T η1,k

1 Tη2,k

2 ...Tηn,kn (x) + ω‖

≤ ‖T η1,k

1 Tη2,k

2 ...Tηn,kn ‖

.‖(T η1,t

1 Tη2,t

2 ...T ηn,tn (ui) − x)‖ + 1

2k

≤ ‖T η1,k

1 Tη2,k

2 ...Tηn,kn ‖ + 1

2ri

≤ 12k

+ 12k

= 1k

Since

‖ω + Tη1,t

1 Tη2,t

2 ...T ηn,tn (ui) − x‖ = ‖ω − x + T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui)‖

≤ ‖ω − x‖ + ‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ui)‖

≤ 12k

+ ‖T η1,t

1 Tη2,t

2 ...T ηn,tn ‖.‖(ui)‖

≤ 12k

+ 12k


= 1k

then h = ω + Tη1,t

1 Tη2,t

2 ...T ηn,tn (ui) ∈ Vk so

‖T η1,t

1 Tη2,t

2 ...T ηn,tn (h)‖ = ‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω + T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui))‖

= ‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω)+T

η1,t

1 Tη2,t

2 ...T ηn,tn (T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui))‖

= ‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω) − ω0+

Tη1,t

1 Tη2,t

2 ...T ηn,tn (T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui)) + ω0‖

≤ ‖T η1,t

1 Tη2,t

2 ...T ηn,tn (ω) − ω0‖+

‖T η1,t

1 Tη2,t

2 ...T ηn,tn (T

η1,t

1 Tη2,t

2 ...T ηn,tn (ui)) + ω0‖

≤ 12k

+ 12k

= 1k

that isT

η1,k

1 Tη2,k

2 ...T ηn,kn (Vk)

⋂Uk �= φ

By this the proof is complete.

Lemma 3.3 Let X be a separable Banach space and T = (T1, T2, ..., Tn) isa chaotic n-tuple of commutative continuous linear operators on X . the tupleT satisfying the hypothesis of The Hypercyclicity Criterion.

proof. Since the tuple T is a chaotic tuple, then have a set of semi-periodic vector, so by the Theorem 3.2, the T satisfying the hypothesis theHypercyclicity Criterion.

Lemma 3.4 Let X be a separable Banach space and T = (T1, T2, ..., Tn) isan n-tuple of commutative bounded linear mapping on X and T have a densegeneralized kernel, then tuple T satisfying the hypothesis of The HypercyclicityCriterion.


proof. For proof of this lemma, it is sufficient to choice a vector x forthe tuple T , by the theorem 3.2 the proof is clear.

ACKNOWLEDGEMENTS. This research was partially supported bya grant from Research Council of Islamic Azad University, Genaveh Branch,so the authors gratefully acknowledge this support.

References

[1] P. S. Bourdon, Orbit of hyponormal operators, Mich. Math. Jour. , 44(1997),345-353.

[2] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spacesof holomorphic functions, Proc. Amer. Math. Soc. , 100 (1987), 281-288.

[3] M. Habibi and F. Safari, n-Tuples and Epsilon Hypercyclicity, Far EastJour. of Math. Sci. , 47 , No. 2 (2010), 219-223.

[4] M. Habibi and F. Safari, Chaoticity of a Pair of Operators, Int. Jour. ofApp. Math. , 24 , No. 2 (2011), 155-160.

[5] M. Habibi and B. Yousefi, Conditions for a tuple of operators to be topo-logically mixing, Int. Jour. of App. Math. , 23 , No. 6 (2010), 973-976.

[6] B. Yousefi and M. Habibi, Syndetically Hypercyclic Pairs, Int. Math. Fo-rum , 5 , No. 66 (2010), 3267 - 3272.

[7] B. Yousefi and M. Habibi, Hereditarily Hypercyclic Pairs, Int. Jour. ofApp. Math. , 24 , No. 2 (2011), 245-249.

[8] B. Yousefi and M. Habibi, Hypercyclicity Criterion for a Pair of WeightedComposition Operators, Int. Jour. of App. Math. , 24 , No. 2 (2011),215-219.

Received: September, 2011

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