This paper introduces the concept of semi-periodic ∞-tuples of commutative bounded linear mappings on a separable Banach space. The authors prove two main results: 1) A hypercyclicity criterion for ∞-tuples, stating conditions under which an ∞-tuple is hypercyclic. 2) If an ∞-tuple is hypercyclic and has a dense generalized kernel, then it satisfies the conditions of the hypercyclicity criterion. The paper aims to expand understanding of dynamical properties of ∞-tuples acting on Banach spaces.

1. Int. Journal of Math. Analysis, Vol. 7, 2013, no. 16, 763 - 766
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Semi-Periodic ∞-Tuples
Kobra Ostad
Department of Mathematics
Dehdasht Branch, Islamic Azad University, Dehdasht, Iran
P.O. Box 7571763111, Dehdasht, Iran
ostad.k@iaudehdasht.ac.ir
Mezban Habibi
Department of Mathematics
Ministry of Education, Molavi School, Stockhom, Sweden
P. O. Box 181 40, Lidingo, Stockholm, Sweden
habibi.m@iaudehdasht.ac.ir
Fatemeh Safari
Department of Mathematics
Dehdasht Branch, Islamic Azad University, Dehdasht, Iran
P. O. Box 7571734494, Dehdasht, Iran
safari.s@iaudehdasht.ac.ir
Abstract
In this paper, we introduce semi-periodic ∞-Tuples of commutative
bounded 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, ...). For
the ∞-tuple T = (T1, T2, ...) the set F = {T1
k1
T2
k2
... : ki ≥ 0, i = 1, 2, ..., n} is
the semigroup generated by T . For x ∈ X take Orb(T , x) = {Sx : S ∈ F} in
other hand Orb(T , x) = {T1
k1
T2
k2
...(x) : ki ≥ 0, i = 1, 2, ...}. The set Orb(T , x)

2. 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) = {T1
k1
T2
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 metric
space with metric d, an element x ∈ X is called fixed point for T if there exist
non-negative integer numbers m1, m2, ... such that Tm1
1 Tm2
2 ...(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 of
numbers. Readers can see [1 − 11] for more information.
2 Preliminary Notes
Semi-Periodic ∞-Tuple. Let X is a Banach space and x ∈ X, the vector
x is called a semi-periodic vector for tuple T = (T1, T2, ...) if the sequence
{Tm1
1 Tm2
2 ...(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 Tμ1
1 Tμ2
2 ...(x) = x. Also
the n-Tuple T = (T1, T2, ...), is called chaotic tuple, if we have tree below
conditions 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 Tα1
1 Tα2
2 ...(U)∩V =
φ
(2). It has a dense set of periodic points, in other word, there is a set X
such that for each x ∈ X, there exist some numbers β1, β2, ... ∈ N such that
Tβ1
1 Tβ2
2 ...(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 be
a 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 strictly
increasing sequences {mj,1}∞
j=1, {mj,2}∞
j=1, ... such that :
1. T
mj,1
1 T
mj,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 T
mj,2
2 ...Sjz → z, on Z as j → ∞,
then T is a hypercyclic ∞-tuple.

3. Semi-periodic ∞-tuples 765
Theorem 3.2 Let X be a separable Banach space and T = (T1, T2, ...) is
an ∞-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 the
hypothesis of The Hypercyclicity Criterion.
proof. Since T is a hypercyclic ∞-tuple, then take the hypercyclic vector
x for T . So the set Orb(T , x) is dense in X , that is
Orb(T , x) = {T1
k1
T2
k2
...(x) : ki ≥ 0, i = 1, 2, ...} = X .
Suppose that F be the generalized kernel of T , that is
M =
k>0
(Ker(T
m1,k
1 T
m2,k
2 ...))
and
N = {x, T
m1,k
1 T
m2,k
2 ...(x), T
m1,k+1
1 T
m2,k+1
2 ...(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
T1
κ1,j
T2
κ2,j
...(νj) → x
as j → ∞. Since
T1
κ1,j
T2
κ2,j
...(νj)
is a hypercyclic vector for the ∞-tuple T so
T1
κ1,j+1
T2
κ2,j+1
...(νj)
also is a hypercyclic vector. By this choice the sequences {η1,j}, {η2,j}, ...} such
that
T1
η1,j
T2
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
T1
η1,j
T2
η2,j
...(ωj) → x

4. 766 K. Ostad, M. Habibi and F. Safari
as j → ∞. Now define Snk
: N ⇒→ X by
Snk
(T1
η1,j
T2
η2,j
...(x)) = T1
η1,j
T2
η2,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 ∞-tuple
T .
ACKNOWLEDGEMENTS. This research was partially supported by a
grant from Research Council of Islamic Azad University, Branch of Dehdasht,
so the authors gratefully acknowledge this support.
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Received: November, 2012