Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 7, 325 - 329HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijcms.2014.416

On Hilbert-Schmidt n-Tuples

Mezban Habibi

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

Iran’s Ministry of Education, Fars Province education organization, Shiraz, IranP. O. Box 7164754818, Shiraz, Iran

Copyright c© 2014 Mezban Habibi. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we study the Hilbert-Schmidt Tuples of operators ona Banach space.

Mathematics Subject Classification: 47A16, 47B37

Keywords: Hypercyclic vector, Hypercyclicity Criterion, Hilbert-Schmidtn-tuple

1 Introduction

Let T1, T2, ..., Tn be commutative bounded linear operators on a Hilbert spaceH, and T = (T1, T2, ..., Tn) be an n-Tuple, put

Γ = {Tm11 Tm2

2 ...Tmnn : m1, m2, ...,mn ≥ 0}

the semigroup generated by T . For x ∈ X , the orbit of x under T is the setOrb(T , x) = {S(x) : S ∈ Γ}, that is

Orb(T , x) = {Tm11 Tm2

2 ...Tmnn (x) : m1, m2, ...,mn ≥ 0}

The vector x is called Hypercyclic vector for T and n-Tuple T is called Hy-percyclic n-Tuple, if the set Orb(T , x) is dense in X , that is

Orb(T , x) = {Tm11 Tm2

2 ...Tmn2 (x) : m1, m2, ...,mn ≥ 0} = X

326 Mezban Habibi

Suppose above assumptions, also let {ai}+∞i=1 and {bj}+∞

j=1 be orthonormal ba-sises in a Hilbert space H. The Tuple T is said to be Hilbert-Schmidt Tuple,if we have

∞∑

i=1

∞∑

j=1

| (T1T2...Tnai, bj) |2<∞

All operators in this paper are commutative operator, reader can see [1–9] formore information.

2 Main Results

Theorem 2.1.[The Hypercyclicity Criterion] Let B be a separable Banachspace and T = (T1, T2, ..., Tn) is an n-tuple of continuous linear mappings onB. If there exist two dense subsets Y and Z in B, and strictly increasingsequences {mj,1}, {mj,2}, ...,{mj,n} such that:1. T

mj,1

1 Tmj,2

2 ...Tmj,nn → 0 on Y as mj,i → ∞ for i = 1, 2, 3, ..., n,

2. There exist function {Sk : Z → B} such that for every z ∈ Z, Skz → 0,and T

mj,1

1 Tmj,2

2 ...Tmj,nn Skz → z,

then T is a Hypercyclic n-tuple.If the tuple T satisfying the hypothesis of previous theorem then we say thatT satisfying the hypothesis of Hypercyclicity criterion.

Theorem 2.2. Suppose X be an F-sequence space whit the unconditionalbasis {eκ}κ∈N . Let T1, T2, ..., Tn are unilateral weighted backward shifts withweight sequence {ai,1 : i ∈ N}, {ai,2 : i ∈ N},..., {ai,n : i ∈ N} and T =(T1, T2, ..., Tn) be an n-tuple of operators T1, T2, ..., Tn. Then the followingassertions are equivalent:(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(∏m

k=1(ak,i)−1em) convergence in X for i = 1, 2, ..., n.

Proof. Proof of the cases (1) → (2) and (2) → (3) are trivial, so we justproof (3) → (4) and (4) → (1). First we proof (3) → (4), for this, Suppose thatT has a non-trivial periodic point, and x = {xn} ∈ X be a non-trivial periodicpoint for T , that is there are μ1, μ2, ..., μn ∈ N such that, T μ1

1 T μ22 ...T μn

n (x) = x.Comparing the entries at positions i+ kMλ for λ = 1, 2, 3, ..., n, k ∈ N ∪ {0},of x and T μ1

1 T μ22 ...T μn

n (x) we find that

xj+kMλ= (

Mλ∏

t=1

(aj+kN+t))xj+(k+1), λ = 1, 2, 3, ..., n

On Hilbert-Schmidt n-Tuples 327

so that we have,

xj+kMλ= (

j+kMλ∏

t=j+1

(at))−1xj = cλ(

j+kMλ∏

t=1

(at))−1, k ∈ N ∪ {0}, λ = 1, 2, 3, ..., n

with

cλ = (j∏

t=1

(mj,λ))xj , λ = 1, 2, 3, ..., n

Since {eκ} is an unconditional basis and x ∈ X it follows that

∞∑

k=0

(1

∏j+kMλt=1 (mj,λ)

)ej+kMλ=

1

cλ

∞∑

k=0

xj+kMλ.ej+kMλ

, λ = 1, 2, 3, ..., n

convergence in X . Without loss of generality, assume that j ≥ N , applyingthe operators T

μ1,j

1 Tμ2,j

2 ...Tμn,jn (x) for j = 1, 2, 3, ...Q− 1, with Q = Min{Mi :

i = 1, 2, ..., n}, to this series and note that Tμ1,j

1 Tμ2,j

2 ...Tμn,jn (en) = ajen−1 for

n ≥ 2 and j = 1, 2, 3, ..., n, we deduce that

∞∑

k=0

(1

∏j+kMλ−υλt=1 (mj,λ)

)ej+kMλ−υλ, λ = 1, 2, 3, ..., n

convergence in X for γ = 0, 1, 2, ..., N − 1. By adding these 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 denseset of periodic points. Since {eκ} is an unconditional basis, condition (4) withproposition 2.3 implies that for each j ∈ N and M,N ∈ N the series

ψ1(j,Mλ) =∞∑

k=0

(1

∏j+kMλt=1 (mk,λ)

)ej+kMλ= (

j∏

t=1

mk,λ).(∞∑

k=0

1∏j+kMλ

t=1 mk,λ

ej+kMλ)

converges for λ = 1, 2, 3, ..., n and define n elements in X . Moreover, if M ≥ ithen T

mjλ,1

1 Tmjλ,2

2 ...Tmjλ,nn = ψλ(jλ,Mλ) for λ = 1, 2, 3, ..., n, and

Tmjλ,1

1 Tmjλ,2

2 ...Tmjλ,nn = ψλ(jλ,Mλ)T

M11 TM2

2 ...TMnn ψλ(jλ,Mλ) = ω(jλ,Mλ

) (1)

Also, if N ≥ ji then

Tmj,1

1 Tmj,2

2 ...Tmj,nn ω((j, i), N) = ω((j, i), N) (2)

for mj,i ≥ N and i = 1, 2, ..., n. So that each ψ(j,N) for j ≤ N is a periodicpoint for T . We shall show that T has a dense set of periodic points. Since{eκ} is a basis, it suffices to show that for every element x ∈ span{eκ : κ ∈ N}

328 Mezban Habibi

there is a periodic point y arbitrarily close to it. For this, let x =∑m

j=1 xjej

and ε > 0. Without lost of generality, for λ = 1, 2, 3, ..., n we can assume that

| xi

i∏

t=1

at,λ |≤ 1 , i = 1, 2, 3, ...,mλ

Since {en} is an unconditional basis, then condition (4) implies that there arean M,N ≥ m such that

‖∞∑

n=Mλ+1

εκ,11

∏κt=1 at,λ

eκ‖ <ε

mλ, λ = 1, 2, 3, ..., n

for every sequences {εκ,i}, i = 1, 2, ..., n taking values 0 or 1. By (1) and (2)the elements

y1 =m1∑

i=1

xiψ(i,Mϕ), ϕ = 1, 2, 3, ..., n

of X is a periodic point for T , and we have‖yλ − x‖ = ‖∑mλ

i=1 xi(ψ(i,Mλ) − ei)‖= ‖∑mλ

i=1(xi∏i

t=1 dt,Mλ)(

∑∞k=1

1∏i+kMλt=1

at,Mλ

ei+kMλ)‖

≤ ∑mλi=1 ‖(xi

∏it=1 dt,Mλ

)(∑∞

k=11∏i+kMλ

t=1at,Mλ

ei+kMλ)‖

≤ ∑mλi=1 ‖(

∑∞k=1

1∏i+kMλt=1

at,Mλ

ei+kMλ)‖

≤ εas λ = 1, 2, ..., n and by this, the proof is complete.

References

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On Hilbert-Schmidt n-Tuples 329

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Received: January 15, 2014