Int. Journal of Math. Analysis, Vol. 7, 2013, no. 16, 763 - 766HIKARI Ltd, www.m-hikari.com

Semi-Periodic ∞-Tuples

Kobra Ostad

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

P.O. Box 7571763111, Dehdasht, Iranostad.k@iaudehdasht.ac.ir

Mezban Habibi

Department of MathematicsMinistry of Education, Molavi School, Stockhom, Sweden

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

Fatemeh Safari

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

P. O. Box 7571734494, Dehdasht, Iransafari.s@iaudehdasht.ac.ir

Abstract

In this paper, we introduce semi-periodic ∞-Tuples of commutativebounded linear mappings on a separable Banach space.

Mathematics Subject Classification: 37A25, 47B37

Keywords: Hypercylicity criterion, ∞-tuple, Hypercylic vector, Semi pe-riodic vector

1 Introduction

Let X be a Banach space and T1, T2, ... are commutative bounded linear map-pings on X . By an ∞-tuple we mean the ∞-component T = (T1, T2, ...). Forthe ∞-tuple T = (T1, T2, ...) the set F = {T1

k1T2k2... : ki ≥ 0, i = 1, 2, ..., n} is

the semigroup generated by T . For x ∈ X take Orb(T , x) = {Sx : S ∈ F} inother hand Orb(T , x) = {T1

k1T2k2...(x) : ki ≥ 0, i = 1, 2, ...}. The set Orb(T , x)

764 K. Ostad, M. Habibi and F. Safari

is called orbit of vector x under T and Tuple T = (T1, T2, ...) is called hyper-cyclic ∞-tuple, if there is a vector x ∈ X such that, the set Orb(T , x) is dense in

X , that is Orb(T , x) = {T1k1T2

k2...(x) : ki ≥ 0, i = 1, 2, ...} = X .. In this case,the vector x is called a hypercyclic vector for the ∞-tuple T . Let X is a metricspace with metric d, an element x ∈ X is called fixed point for T if there existnon-negative integer numbers m1, m2, ... such that Tm1

1 Tm22 ...(x) = x. Fixed

point for a functional is defining similarly. Let X is a Banach space and x ∈ X ,the vector x is called a semi-periodic vector for T = (T1, T2, ...) if the sequence

{T (m1,k)1 T

(m2,k)2 ...(x)} be semi-compact. In this case T is called semi-periodic

∞-tuple. All of operators in this paper are commutative bounded linear oper-ators on a Banach space. Also, note that by {j, i} or (j, i) we mean a number,that was showed by this mark and related with this indexes, not a pair ofnumbers. Readers can see [1 − 11] for more information.

2 Preliminary Notes

Semi-Periodic ∞-Tuple. Let X is a Banach space and x ∈ X, the vectorx is called a semi-periodic vector for tuple T = (T1, T2, ...) if the sequence{T m1

1 Tm22 ...(x)} be semi-compact. In this case T is called semi-periodic tuple.

The vector x in X is called a Periodic vector for the n-Tuple T = (T1, T2, ...),if there exist some numbers μ1, μ2, ... ∈ N such that T μ1

1 T μ22 ...(x) = x. Also

the n-Tuple T = (T1, T2, ...), is called chaotic tuple, if we have tree belowconditions together,(1). It is topologically transitive, that is, if for any given open sets U and V,there exist positive integer numbers α1, α2, ... ∈ N such that T α1

1 T α22 ...(U)∩V �=

φ(2). It has a dense set of periodic points, in other word, there is a set Xsuch that for each x ∈ X , there exist some numbers β1, β2, ... ∈ N such thatT β1

1 T β22 ...(x) = x

(3). It has a certain property called sensitive dependence on initial conditions.

3 Main Results

Theorem 3.1 (The Hypercyclicity Criterion for ∞-Tuples) Let X bea separable Banach space and T = (T1, T2, ...) is an ∞-tuple of continuous lin-ear mappings on X . If there exist two dense subsets Y and Z in X , and strictlyincreasing sequences {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 ...Sjz → z, on Z as j → ∞,then T is a hypercyclic ∞-tuple.

Semi-periodic ∞-tuples 765

Theorem 3.2 Let X be a separable Banach space and T = (T1, T2, ...) isan ∞-tuple of commutative bounded linear mapping on X . If T is a hyper-cyclic tuple and it have a dense generalized kernel, then tuple T satisfying thehypothesis of The Hypercyclicity Criterion.

proof. Since T is a hypercyclic ∞-tuple, then take the hypercyclic vectorx for T . So the set Orb(T , x) is dense in X , that is

Orb(T , x) = {T1k1T2

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

Suppose that F be the generalized kernel of T , that is

M =⋃

k>0

(Ker(Tm1,k

1 Tm2,k

2 ...))

and

N = {x, Tm1,k

1 Tm2,k

2 ...(x), Tm1,k+11 T

m2,k+12 ...(x)), ...}.

Since M = X and x is a hypercyclic vector for T , then

N = X (1)

Now we can take increasing sequences of positive integers {κ1,j}, {κ2,j}, ...}and {νj} such that νj → 0 and

T1κ1,jT2

κ2,j ...(νj) → x

as j → ∞. Since

T1κ1,jT2

κ2,j ...(νj)

is a hypercyclic vector for the ∞-tuple T so

T1κ1,j+1T2

κ2,j+1...(νj)

also is a hypercyclic vector. By this choice the sequences {η1,j}, {η2,j}, ...} suchthat

T1η1,jT2

η2,j ...(x) → x

and

{κ1,t} > {κ1,j}, {κ2,t} > {κ2,j}, ...} > {κn,j}

as j → ∞. So, there are {μ1,j}, {μ2,j}, ...} such that

κ1,j + μ1,j = η1,j, κ2,j + μ2,j = η2,j , ...

So

T1η1,jT2

η2,j ...(ωj) → x

766 K. Ostad, M. Habibi and F. Safari

as j → ∞. Now define Snk: N ⇒→ X by

Snk(T1

η1,jT2η2,j ...(x)) = T1

η1,jT2η2,j ...(ωk)

so that η1,j = 0, 1, 2, ... as j = 1, 2, .... Now we have M and N and Snk: N →

X that satisfying the hypothesis of Hypercyclicity Criterion for the ∞-tupleT .

ACKNOWLEDGEMENTS. This research was partially supported by agrant from Research Council of Islamic Azad University, Branch of Dehdasht,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] Mezban Habibi, On Syndetically Hypercyclic Tuples, International Math-ematical Forum, 7, No. 52(2012), 2597 - 2602.

[4] Mezban Habibi, N-Tuples and Chaoticity, Int. Journal of Math. Analysis,6, No. 14(2012), 651 - 657.

[5] Mezban Habibi, ∞-Tuples of Bounded Linear Operators on Banach Space,International Mathematical Forum, 7, No. 18(2012), 861 - 866.

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

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

[8] 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.

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

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

[11] 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: November, 2012