Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 41, 1993 - 1998

∞-Tuples of Operators and Hereditarily

Mezban Habibi

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

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

Sadegh Masoudi

Young reserch groupIslamic Azad University, Dehdasht beranch, Dehdasht, Iran

masoudi34@gmail.com

Fatemeh Safari

Department of MathematicsIslamic Azad University, Branch of Dehdasht, Dehdasht, Iran

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

Abstract

In this paper, we introduce for an ∞-tuple of operators on a Banachspace and some conditions to an ∞-tuple to satisfying the HypercyclicityCriterion.

Mathematics Subject Classification: 37A25, 47B37

Keywords: Hypercylicity criterion, ∞-tuple, Hypercyclic vector, Heredi-tarily ∞-tuple

1 Introduction

Let X be an infinite dimensional Banach space and T1, T2, ... are commutativebounded linear operators on X . By an ∞-tuple we mean the ∞-componentT = (T1, T2, ...). For the ∞-tuple T = (T1, T2, ...) the set

F = {T1k1T2

k2 ... : ki ≥ 0, i = 1, 2, ..., n}

1994 M. Habibi, S. Masoudi, F. Safari

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

Orb(T , x) = {Sx : S ∈ F}.

In other hand

Orb(T , x) = {T1k1T2

k2 ...(x) : ki ≥ 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, 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 .

2 Preliminary Notes / Materials and Methods

Definition 2.1 Let V be a topological vector space(TV S) and T1, T2, ... arebounded linear mapping on V, and T = (T1, T2, ...) be an ∞-tuple of operators.The ∞-tuple T is called weakly mixing if

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

is topologically transitive.

Definition 2.2 Let

{m(k,1)}∞k=1, {m(k,2)}∞k=1, ...

be increasing sequences of non-negative integers. The ∞-tuple T = (T1, T2, ...)is called hereditarily hypercyclic with respect to

{mj,1}∞j=1, {mj,2}∞j=1, ...

if for all subsequences{m′

j,1}∞j=1, {m′j,2}∞j=1, ...

of{mj,1}∞j=1, {mj,2}∞j=1, ...

respectively, the sequence

{Tm′

(k,1)

1 Tm′

(k,2)

2 ...}is hypercyclic. In the other hand, there exists a vector x in X such that

{Tm′

(k,1)

1 Tm′

(k,2)

2 ...(x)} = X .

∞-Tuples of operators and hereditarily 1995

Note 2.3 Note that, if X be an finite dimensional Banach space, then thereare no hypercyclic operator on X , also there are no ∞-tuple or n-tuple on X .

All of operators in this paper are commutative bounded linear operatorson a Banach space. Also, note that by {j, i} or (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 − 10] for some information.

3 Results and Discussion

These are the main results of the paper.If the ∞-tuple of continuous linear mappings satisfying hypothesis of bellow

theorem then we say that is satisfying the Hypercyclicity Criterion.

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.

Now, the main theorem of this paper is the bellow theorem.

Theorem 3.2 An ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic withrespect to increasing sequences of non-negative integers

{mj,1}∞j=1, {mj,2}∞j=1, ...

if and only if for all given any two open sets U , V, there exist some positiveintegers M1, M2, ... such that

Tmk,1

1 Tmk,2

2 ...(U)⋂

V �= φ

for

∀mk,1 > M1, ∀mk,2 > M2, ...

Proof. Let T = (T1, T2, ...) be hereditarily hypercyclic ∞-tuple with respectto increasing sequences of non-negative integers

{mj,1}∞j=1, {mj,2}∞j=1, ...

1996 M. Habibi, S. Masoudi, F. Safari

and suppose that there exist some open sets U , V such that

Tmk,1

1 Tmk,2

2 ...(U)⋂

V = φ

for some subsequence

{m′j,1}∞j=1, {m′

j,2}∞j=1, ...

of

{mj,1}∞j=1, {mj,2}∞j=1, ...

respectively. Since the ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic withrespect to

{mj,1}∞j=1, {mj,2}∞j=1, ...

, thus

{T m′k,1

1 Tm′

k,2

2 ...}

is hypercyclic, and so we get a contradiction.Conversely, suppose that

{m′j,1}∞j=1, {m′

j,2}∞j=1, ...

are arbitrary subsequences of

{mj,1}∞j=1, {mj,2}∞j=1, ...

respectively, and U , V are open sets in X , satisfying

Tmk,1

1 Tmk,2

2 ...(U)⋂

V �= φ

for any

m(k,j) > Mj , j = 1, 2, ...

. So there exist (i, j), large enough for j = 1, 2, ... such that m(ki,j) > Mj forj = 1, 2, ... and

Tm(ki,1)

1 Tm(ki,2)

2 ...(U)⋂

V �= φ.

This implies that

{T m(ki,1)

1 Tm(ki,2)

2 ...}

is hypercyclic, so the ∞-tuple T = (T1, T2, ..., Tn) is indeed hereditarily hyper-cyclic with respect to the sequences

{m(k,1)}∞k=1, {m(k,2)}∞k=1, ....

By this the proof is complete.

∞-Tuples of operators and hereditarily 1997

Theorem 3.3 An ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic withrespect to increasing sequences of non-negative integers {mj,1}∞j=1, {mj,2}∞j=1, ...,if and only if for all given any two open sets U , V, there exist some positive in-tegers M1, M2, ... such that T

mk,1

1 Tmk,2

2 ...(U)⋂V �= φ for ∀mk,1 > M1, ∀mk,2 >

M2, ....

Proof. Let T = (T1, T2, ...) be hereditarily hypercyclic ∞-tuple with respectto increasing sequences of non-negative integers {mj,1}∞j=1, {mj,2}∞j=1, ..., and

suppose that there exist some open sets U , V such that Tmk,1

1 Tmk,2

2 ...(U)⋂V =

φ for some subsequence {m′j,1}∞j=1, {m′

j,2}∞j=1, ... of {mj,1}∞j=1, {mj,2}∞j=1, ... re-spectively. Since the ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic with

respect to {mj,1}∞j=1, {mj,2}∞j=1, ..., thus {T m′k,1

1 Tm′

k,2

2 ...} is hypercyclic, and sowe get a contradiction.Conversely, suppose that {m′

j,1}∞j=1, {m′j,2}∞j=1, ... are arbitrary subsequences

of {mj,1}∞j=1, {mj,2}∞j=1, ... respectively, and U , V are open sets in X , satisfying

Tmk,1

1 Tmk,2

2 ...(U)⋂

V �= φ

for any m(k,j) > Mj , j = 1, 2, .... So there exist (i, j), large enough for j =1, 2, ... such that m(ki,j) > Mj for j = 1, 2, ... and

Tm(ki,1)

1 Tm(ki,2)

2 ...(U)⋂

V �= φ.

This implies that {T m(ki,1)

1 Tm(ki,2)

2 ...} is hypercyclic, so the ∞-tuple T = (T1, T2, ..., Tn)is indeed hereditarily hypercyclic with respect to the sequences {m(k,1)}∞k=1, {m(k,2)}∞k=1, ....By this the proof is complete.

References

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

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

[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.

1998 M. Habibi, S. Masoudi, F. Safari

[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.

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

[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 App. Math. , 24 (2) (2011), 215-219.

Received: May, 2012