International Journal of Applied Mathematics————————————————————–Volume 24 No. 2 2011, 155-160

CHAOTICITY OF A PAIR OF OPERATORS

Mezban Habibi1 §, Fatemeh Safari2

1,2Department of MathematicsIslamic Azad University

Branch of DehdashtP.O. Box 7571763111, Dehdasht, IRAN

1e-mail: habibi.m@iaudehdasht.ac.ir2e-mail: safari.s@iaudehdasht.ac.ir

Abstract: In this paper we characterize the Condition for Chaoticity of apair of operators on a Frechet space.

AMS Subject Classification: 47A16, 37C25, 47B37Key Words: hypercyclic vector, hypercyclicity criterion, periodic vector,chaotic pair, pair of operators

1. Introduction

Let X be a Frechet space and T1 , T2 be two bounded linear operators on Xand T = (T1,T2) be a pair of operators T1,T2. Put Orb(T , x) = {Tm

1 T n2 (x) :

m,n ≥ 0)}. The set Orb(T , x) is called orbit of vector x under T . The pairT = (T1, T2) is called hypercyclic pair, if there exists a vector x in X such that,the Orbit of x under T is dense in X , that is Orb(T , x) = X . The vector x in Xis called a Periodic vector for the pair T = (T1, T2), if there exist some numbersm,n ∈ N such that Tm

1 Tn2 (x) = x. Also the pair T = (T1, T2) is called chaotic

pair, if we have the following 3 conditions satisfied:

Received: August 11, 2010 c© 2011 Academic Publications§Correspondence author

156 M. Habibi, F. Safari

(1) It is topologically transitive, that is, if for any given open sets U and V,there exist two positive integer numbers m,n such that, Tm

1 T n2 (U) ∩ V �= φ.

(2) It has a dense set of periodic points, that is, for each x ∈ X , there existsome numbers m,n ∈ N such that Tm

1 T n2 (x) = x.

(3) It has a certain property called sensitive dependence on initial condi-tions.

2. Main Results

A nice criterion, namely the Hypercyclicity Criterion, is used in the proof ofour main theorem.

Theorem 2.1. (The Hypercyclicity Criterion) Suppose X is a separableBanach space and T = (T1, T2) is a pair of continuous linear mapping on F . Ifthere exist two dense subsets Y and Z in X and two strictly increasing sequences{mj} and {nj} such that:

1. Tmj

1 Tnj

2 → 0,

2. There exist function {Sj : Z → F} such that for every z ∈ Z, Sjz → 0,and T

mj

1 Tnj

2 Sjz → z,

then T is a hypercyclic pair.

Theorem 2.2. Suppose X is an F-sequence space over set I. Consider thefollowing assertions:

(1). {ei}i∈I is an unconditional basis for X ,

(2). {ei}i∈I is a basis in some ordering and if {xi} ∈ X , then also {εixi} ∈ Xwhenever each εi is either 0 or 1,

(3). {ei}i∈I is a basis in some ordering and if {xi} ∈ X then also {cixi} ∈ Xwhenever each ci is a bounded sequence of scalar.

Then we have the following implications: (1) ⇔ (2) ⇐ (3). If X is in addi-tion locally convex or locally bounded, then all three assertions are equivalent(Grosse Erdmann).

Remark 2.3. A sequence space that satisfies the second condition in (2)is called Monotone, and one that satisfies the second condition in (3) is calledsolid.

CHAOTICITY OF A PAIR OF OPERATORS 157

Theorem 2.4. Suppose X is an F-sequence space with the unconditionalbasis {en}n∈N . Let T1, T2 be unilateral weighted backward shifts with weightsequence {ai : i ∈ N} and {bl : l ∈ N}. Then the following assertions areequivalent:

(1). T is chaotic.

(2). T is hypercyclic and has a non-trivial periodic point.

(3). T has a non-trivial periodic point.

(4). the series Σ∞m=1(

∏mk=1(ak)−1em) and Σ∞

n=1(∏n

l=1(bl)−1en) convergence

in X .

Proof. Proof of the cases (1) → (2) and (2) → (3) are trivial, so we onlyprove (3) → (4) and (4) → (1).

First we proof (3) → (4). For this, suppose that T has a non-trivial periodicpoint, and x = {xn} ∈ X is a non-trivial periodic point for T , that is there areM and N and j ∈ N such that, TM

1 TN2 (x) = x and xj �= 0. Comparing the

entries at positions j + kN , and j + kN , k ∈ N ∪ {0}, of x and TM1 TN

2 (x), wefind that

xj+kM = (M∏

t=1

(aj+kN+t))xj+(k+1)

xj+kN = (N∏

t=1

(bj+kN+t))xj+(k+1)

so that we have,

xj+kM = (j+kM∏

t=j+1

(at))−1xj = c(j+kM∏

t=1

(at))−1, k ∈ N ∪ {0}

xj+kN = (j+kN∏

t=j+1

(bt))−1xj = d(j+kN∏

t=1

(bt))−1, k ∈ N ∪ {0}

with c = (∏j

t=1(at))xj and d = (∏j

t=1(bt))xj . Since {en} is an unconditionalbasis and x ∈ X , it follows that

∞∑

k=0

(1

∏j+kMt=1 (at)

)ej+kM =1c

∞∑

k=0

xj+kM .ej+kM ,

158 M. Habibi, F. Safari

∞∑

k=0

(1

∏j+kNt=1 (bt)

)ej+kN =1d

∞∑

k=0

xj+kN .ej+kN ,

with convergence in X . Without loss of generality we may assume that j ≥ N .Applying the operators T , T 2, T 3,..., TR−1, with R = min{M,N}, to thisseries and note that T1(en) = anen−1 and T2(en) = bnen−1 for n ≥ 2, we deducethat ∞∑

k=0

(1

∏j+kM−υt=1 (at)

)ej+kM−υ,

∞∑

k=0

(1

∏j+kN−νt=1 (bt)

)ej+kN−ν,

convergent in X for γ = 0, 1, 2, ..., N − 1. Adding these N series, we see thatcondition (4) holds.

Proof of (4) ⇒ (1). It follows from Theorem 2.1, that under condition (4)the operator T is hypercyclic. Hence it remains to show that T has a dense setof periodic points. Since {en} is an unconditional basis, condition (4) impliesthat for each j ∈ N and M,N ∈ N the series

ψ(j,M) =∞∑

k=0

(1

∏j+kMt=1 (at)

)ej+kM = (j∏

t=1

at).(∞∑

k=0

1∏j+kM

t=1 at

ej+kM),

ω(j,N) =∞∑

k=0

(1

∏j+kNt=1 (bt)

)ej+kN = (j∏

t=1

bt).(∞∑

k=0

1∏j+kN

t=1 btej+kN)

are convergent and define two elements in X . Moreover, if M ≥ i, then

TM1 TN

2 ψ(j,M) = ψ(j,M) (2.1)

and, if N ≥ j thenTM

1 TN2 ω(j,N) = ω(j,N) (2.2)

so that each ψ(j,N) for j ≤ N is a periodic point for T . We shall show that Thas a dense set of periodic points. Since {en} is a basis, it suffices to show thatfor every element x ∈ span{en : n ∈ N} there is a periodic point y arbitrarilyclose to it. For this, let x =

∑mj=1 xjej and ε > 0. We can assume without lost

of generality that

| xi

i∏

t=1

at |≤ 1 , i = 1, 2, 3, ...,m,

CHAOTICITY OF A PAIR OF OPERATORS 159

| xj

j∏

t=1

bt |≤ 1 , j = 1, 2, 3, ...,m.

Since {en} is an unconditional basis, then condition (4), Theorem 2.4, impliesthat there are an M,N ≥ m such that

‖∞∑

n=M+1

εn1∏n

t=1 aten‖ <

ε

m,

‖∞∑

n=N+1

ηn1∏n

t=1 bten‖ <

η

m,

for every sequences {εn} and {ηn} taking values 0 or 1. By (2.1) and (2.2) theelements

y1 =m∑

i=1

xiψ(i,M) , y2 =m∑

j=1

xjω(j,N)

of X are periodic points for T , and we have

‖y1 − x‖ = ‖m∑

i=1

xi(ψ(i,M) − ei)‖ = ‖m∑

i=1

(xi

i∏

t=1

dt)(∞∑

k=1

1∏i+kM

t=1 at

ei+kM )‖

≤m∑

i=1

‖(xi

i∏

t=1

dt)(∞∑

k=1

1∏i+kM

t=1 at

ei+kM )‖ ≤m∑

i=1

‖(∞∑

k=1

1∏i+kM

t=1 at

ei+kM )‖ ≤ ε

and

‖y2 − x‖ = ‖m∑

j=1

xj(ω(j,N) − ej)‖ = ‖m∑

j=1

(xj

j∏

t=1

dt)(∞∑

k=1

1∏j+kN

t=1 btej+kN )‖

≤m∑

j=1

‖(xj

j∏

t=1

dt)(∞∑

k=1

1∏j+kN

t=1 btej+kN)‖ ≤

m∑

j=1

‖(∞∑

k=1

1∏j+kN

t=1 btej+kN )‖ ≤ ε.

By this, the proof is complete.

Acknowledgments

This research was partially supported by a grant from Research Council ofIslamic Azad University, Branch of Dehdasht, so the authors gratefully ac-knowledge this support.

160 M. Habibi, F. Safari

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