This document presents results on the hypercyclicity of tuples of commutative bounded linear operators on Banach spaces. It defines what it means for an infinite tuple of operators T to be hypercyclic or epsilon-hypercyclic. It proves that if T is epsilon-hypercyclic for every epsilon greater than 0, then T is hypercyclic. It also proves the Hypercyclicity Criterion, stating that if T satisfies two conditions involving dense subsets, then T is hypercyclic. The paper studies hypercyclic tuples to further the understanding of hypercyclic operators.

1. Pure Mathematical Sciences, Vol. 3, 2014, no. 1, 17 - 21
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/pms.2014.359
On Hypercyclicity ∞-Tuples
of Commutative Bounded Linear Operators
S. Nasrin Hoseini M.
Department of Mathematics
Genaveh Branch, Islamic Azad University, Genaveh, Iran
P. O. Box 7561738455, Borazjan, Iran
Mezban Habibi
Department of Mathematics
Ministry of Education, Fars province organization, Shiraz, Iran
P. O. Box 181 56, Svalsvagen 1, Lidingo, Stockholm, Sweden
Copyright c 2014 S. Nasrin Hoseini M. and Mezban Habibi. This is an open access
article distributed under the Creative Commons Attribution License, which permits unre-
stricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Abstract
In this paper we study the epsilon hypercyclicity on ∞-tuple makes
with commutative bounded linear operators on Banach spaces.
Mathematics Subject Classification: 47A16, 37A25
Keywords: Hypercyclic vector, ∞-tuple, Hypercyclicity Criterion, Ba-
nach space, Epsilon hypercyclicity
1 Introduction
If T1, T2, ..., Tn, ... be commutative bounded linear mappings on a Banach space
X and T = (T1, T2, ..., Tn, ...), by T (x) or Tm1
1 Tm2
2 ...Tmn
n ...(x) we mean
Supn→∞{Tm1
1 Tm2
2 ...Tmn
n (x) : m1, m2, ..., mn ≥ 0}
So
T (x) = Tm1
1 Tm2
2 ...Tmn
n ...(x)
= Supn→∞{Tm1
1 Tm2
2 ...Tmn
n (x) : m1, m2, ..., mn ≥ 0}

2. 18 S. Nasrin Hoseini M. and Mezban Habibi
Definition 1.1 Let T1, T2, ..., Tn, ... be commutative bounded linear opera-
tors on a Banach space X . For ∞-tuple T = (T1, T2, ..., Tn, ...), put
Γ = {Tm1
1 Tm2
2 ...Tmn
n ... : m1, m2, ..., mn, ... ≥ 0}
For x ∈ X, the orbit of x under T is the set Orb(T , x) = {S(x) : S ∈ Γ}, that
is
Orb(T , x) = {Tm1
1 Tm2
2 ...Tmn
n ...(x) : m1, m2, ..., mn... ≥ 0}
The vector x is called hypercyclic vector for T and ∞-tuple T is called hyper-
cyclic ∞-tuple, if the set Orb(T , x) is dense in X , that is
Orb(T , x) = {Tm1
1 Tm2
2 ...Tmn
n ...(x) : m1, m2, ..., mn... ≥ 0} = X
Also if be a number in (0, 1) and x be a vector of X . The vector x is called
-hypercyclic vector for ∞-tuple T = (T1, T2, ..., Tn, ...) and the ∞-tuple T is
called -hypercyclic ∞-tuple, if for every non zero vector y ∈ X, there exist
some integers m1, m2, ..., mn, ... such that
Tm1
1 Tm2
2 ...Tmn
2 ...x − y < y
All operators in this paper are commutative operators on a Banach space X .
Readers can see [1 − −10] for more information.
2 Main Results
The bellow theorem namely the Hypercyclicity Criterion is useful theorem in
operator theory, that it used in papular theorem’s proof, if ∞-tuple T satisfy
this theorem then we say that it satisfying the The Hypercyclicity Criterion.
Theorem 2.1 (The Hypercyclicity Criterion) Let X be a separable Ba-
nach space and T = (T1, T2, ..., Tn, ...) is an ∞-tuple of continuous linear map-
pings on X . If there exist two dense subsets Y and Z in X , and n strictly
increasing sequences {mj,1}, {mj,2}, ..., {mj,n}, ... such that :
1. T
mj,1
1 T
mj,2
2 ...T
mj,n
n ... → 0 on Y as j → ∞,
2. There exist function {Sj : Z → X} such that for every z ∈ Z, Sjz → 0,
and T
mj,1
1 T
mj,2
2 ...T
mj,n
n ...Sjz → z,
then T is a hypercyclic ∞-tuple.
Theorem 2.2 Let X be a separable Hilbert space and T = (T1, T2, ..., Tn, ...)
be an ∞-tuple of commutative bounded linear operators on a Hilbert space X .
If for every > 0, the ∞-tuple T is -hypercyclic, then T is a hypercyclic .

3. On hypercyclicity ∞-tuples of commutative bounded linear operators 19
Proof . Note that, if T is a Hypercyclic ∞-tuple, then σp(T∗
) = φ, (T ∗
=
(T∗
1 , T∗
2 , ..., T∗
n , ...)), also all spaces that admitted some hypercyclic operator,
are infinite dimensional spaces, so we can assume that X be infinite dimensional
space. Suppose that U and V are subset of X . Give u ∈ U, v ∈ V two
nonzero element and δ > 0 so large that B(u, δ) ⊂ U and B(v, δ) ⊂ V so that
δ < Max{ u , v }. Take x such that x be an -hypercyclic for T with
property < δ
6Max{ u , v }
, then there exist m1,0, m2,0, ..., mn,0, ... such that
T
m1,0
1 T
m2,0
2 ...Tmn,0
n ...x − u < u < δ
thus we have
Tm1
1 Tm2
2 ...Tmn
n ...x ∈ U
Suppose on the contrary that there are only finitely many such integers
m1,1, m2,1, . . . , mn,1
m1,2, m2,2, . . . , mn,2
. . .
m1,t, m2,t, . . ., mn,t
. . .
As above, for each u ∈ X with
u − u <
2δ
3
there exist integers
m1(u ), m2(u ), ..., mn(u ), ...
which satisfies
T
m1(u )
1 T
m2(u )
2 ...Tmn(u )
n ...x − u ≤ u 2 u <
δ
3
Since
T
m1(u )
1 T
m2(u )
2 ...Tmn(u )
n ...x−u ≤ T
m1(u )
1 T
m2(u )
2 ...Tmn(u )
n ...x−u + u−u < δ
we have
mk(u ) ∈ {m1,k, m2,k, ..., mn,k, ...}
for k = 1, 2, ..., t, ... and the ball B(u, 2δ
3
) is covered by a finite number balls
B(T
m1,1
1 T
m2,1
2 ...Tmn,1
n x,
δ
3
, ...)

4. 20 S. Nasrin Hoseini M. and Mezban Habibi
B(T
m1,2
1 T
m2,2
2 ...Tmn,2
n x,
δ
3
, ...)
. . .
B(T
m1,t
1 T
m2,t
2 ...Tmn,t
n x,
δ
3
, ...)
. . .
Thus in an infinite dimensional space this is impossible. So there are infinitely
many integers as m1, m2, . . . , mn, ... with
Tm1
1 Tm2
2 ...Tmn
n ...x − u < δ
Then there exist mi,k > mi,0 for k = 1, 2, ..., t, ... and i = 1, 2, ..., n, ... such that
Tm1
1 Tm2
2 ...Tmn
n ...x ∈ V
Thus
T
m1,1−m1,0
1 T
m2,2−m2,0
2 ...Tmn,n−mn,0
n ...T
m1,0
1 T
m2,0
2 ...Tmn,0
n ...x
is belong to
V T
m1,1−m1,0
1 T
m2,2−m2,0
2 ...Tmn,n−mn,0
n ...(U)
that is
T
m1,1
1 T
m2,2
2 ...Tmn,n
n ...x ∈ (V T
m1,1−m1,0
1 T
m2,2−m2,0
2 ...Tmn,n−mn,0
n ...)
Here it can be concluded that T is hypercyclic ∞-tuple.
References
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5. On hypercyclicity ∞-tuples of commutative bounded linear operators 21
[7] B. Yousefi and M. Habibi, Syndetically Hypercyclic Pairs, Int. Math. Fo-
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Received: May 1, 2013