1. Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 41, 1993 - 1998
∞-Tuples of Operators and Hereditarily
Mezban Habibi
Department of Mathematics
Dehdasht Branch, Islamic Azad University, Dehdasht, Iran
P. O. Box 181 40, Lidingo, Stockholm, Sweden
habibi.m@iaudehdasht.ac.ir
Sadegh Masoudi
Young reserch group
Islamic Azad University, Dehdasht beranch, Dehdasht, Iran
masoudi34@gmail.com
Fatemeh Safari
Department of Mathematics
Islamic Azad University, Branch of Dehdasht, Dehdasht, Iran
P. O. Box 7571734494, Dehdasht, Iran
safari.s@iaudehdasht.ac.ir
Abstract
In this paper, we introduce for an ∞-tuple of operators on a Banach
space and some conditions to an ∞-tuple to satisfying the Hypercyclicity
Criterion.
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 commutative
bounded linear operators 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}

2. 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) = {T1
k1
T2
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) = {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 .
2 Preliminary Notes / Materials and Methods
Definition 2.1 Let V be a topological vector space(TV S) and T1, T2, ... are
bounded 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
{mj,1}∞
j=1, {mj,2}∞
j=1, ...
of
{mj,1}∞
j=1, {mj,2}∞
j=1, ...
respectively, the sequence
{T
m(k,1)
1 T
m(k,2)
2 ...}
is hypercyclic. In the other hand, there exists a vector x in X such that
{T
m(k,1)
1 T
m(k,2)
2 ...(x)} = X .

3. ∞-Tuples of operators and hereditarily 1995
Note 2.3 Note that, if X be an finite dimensional Banach space, then there
are 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 operators
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 − 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 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.
Now, the main theorem of this paper is the bellow theorem.
Theorem 3.2 An ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic with
respect 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
integers M1, M2, ... such that
T
mk,1
1 T
mk,2
2 ...(U) V = φ
for
∀mk,1 > M1, ∀mk,2 > M2, ...
Proof. Let T = (T1, T2, ...) be hereditarily hypercyclic ∞-tuple with respect
to increasing sequences of non-negative integers
{mj,1}∞
j=1, {mj,2}∞
j=1, ...

4. 1996 M. Habibi, S. Masoudi, F. Safari
and suppose that there exist some open sets U, V such that
T
mk,1
1 T
mk,2
2 ...(U) V = φ
for some subsequence
{mj,1}∞
j=1, {mj,2}∞
j=1, ...
of
{mj,1}∞
j=1, {mj,2}∞
j=1, ...
respectively. Since the ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic with
respect to
{mj,1}∞
j=1, {mj,2}∞
j=1, ...
, thus
{T
mk,1
1 T
mk,2
2 ...}
is hypercyclic, and so we get a contradiction.
Conversely, suppose that
{mj,1}∞
j=1, {mj,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
T
mk,1
1 T
mk,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
T
m(ki,1)
1 T
m(ki,2)
2 ...(U) V = φ.
This implies that
{T
m(ki,1)
1 T
m(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.

5. ∞-Tuples of operators and hereditarily 1997
Theorem 3.3 An ∞-tuple T = (T1, T2, ...) is hereditarily hypercyclic with
respect 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 T
mk,2
2 ...(U) V = φ for ∀mk,1 > M1, ∀mk,2 >
M2, ....
Proof. Let T = (T1, T2, ...) be hereditarily hypercyclic ∞-tuple with respect
to 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 T
mk,1
1 T
mk,2
2 ...(U) V =
φ for some subsequence {mj,1}∞
j=1, {mj,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
mk,1
1 T
mk,2
2 ...} is hypercyclic, and so
we get a contradiction.
Conversely, suppose that {mj,1}∞
j=1, {mj,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
T
mk,1
1 T
mk,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
T
m(ki,1)
1 T
m(ki,2)
2 ...(U) V = φ.
This implies that {T
m(ki,1)
1 T
m(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.
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Received: May, 2012