1. International Mathematical Forum, Vol. 7, 2012, no. 52, 2597 - 2602
On Syndetically Hypercyclic Tuples
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
Abstract
In this paper we will give some conditions for a tuple of operators
or tuple of weighted shifts to be Syndetically Hypercyclic.
Mathematics Subject Classification: 47A16, 47B37
Keywords: Syndetically tuple, Hypercyclic tuple, Hypercyclic vector, Hy-
percyclicity Criterion, Syndetically Hypercyclic
1 Introduction
Let F be a topological vector space(TV S) and T1,T2,...,Tn are continuous map-
ping on F, and T = (T1, T2, ..., Tn) be a tuple of operators T1,T2,...,Tn.The
tuple T is weakly mixing, if and only if, For any pair of non-empty open sub-
sets U,V in X, and for any syndetic sequences {mk,1}, {mk,2}, ..., {mk,n} with
Supk(nk+1,j − nk,j) < ∞ for j = 1, 2, ..., n, then there exist mk,1, mk,2, ..., mk,n
such that T
mk,1
1 T
mk,2
2 ...T
mk,n
n U ∩ V = ∅, also, if and only if, it suffices in pre-
vious condition, to consider only those sequences mk,1, mk,2, ..., mk,n for which
there is some m1 ≥ 1, m2 ≥ 1,...,mn ≥ 1 with mk,1 ∈ {mj, 2mj} for all k and
all j. Reader can see [1 − −10] for some information.
2 Main Results
Let X be a metrizable and complete topological vector space(F−space) and
T = (T1, T2, ..., Tn) is an n-tuple of operators, then we will let
F = {T1
k1
T2
k2
...Tn
kn
: ki ≥ 0}
be the semigroup generated by T . For x ∈ X we take
Orb(T , x) = {Sx : S ∈ F} = {T1
k1
T2
k2
...Tn
kn
(x) : ki ≥ 0, i = 1, 2, ..., n}.

2. 2598 M. Habibi
The set Orb(T , x) is called, orbit of vector x under T and Tuple T = (T1, T2, ..., Tn)
is called hypercyclic pair if the set Orb(T , x) is dense in X , that is
Orb(T , x) = {T1
k1
T2
k2
...Tn
kn
(x) : ki ≥ 0, i = 1, 2, ..., n} = X .
The tuple T = (T1, T2, ..., Tn) is called topologically mixing if for any given
open subsets U and V of X , there exist positive numbers K1, K2,...,Kn such
that
T1
k1,i
T2
k2,i
...Tn,i
kn,i
(U) ∩ V = φ , ∀kj,i ≥ Ki , ∀j = 1, 2, ..., n
A sequence of operators {Tn}n≥0 is said to be a hypercyclic sequence on X if
there exists some x ∈ X such that its orbit is dense in X, that is
Orb({Tn}n≥0, x) = Orb({x, T1x, T2x, ...} = X
In this case the vector x is called hypercyclic vector for the sequence {Tn}n≥0.
Note that, if {Tn}n≥0 is a hypercyclic sequence of operators on X, then X
is necessarily separable. Also note that, the sequence {Tn
} is a hypercyclic
sequence on X, if and only if, the operator T is hypercyclic operator on X.
T is said to satisfy the Hypercyclicity Criterion if it satisfies the hypothesis of
below theorem.
Theorem 2.1 (The Hypercyclicity Criterion) Let X be a separable Ba-
nach space and T = (T1, T2, ..., Tn) is an n-tuple of continuous linear 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,...,{mj,n}∞
j=1 such that:
1. T
mj,1
1 T
mj,2
2 ...T
mj,n
n → 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 ...T
mj,n
n Sjz → z,
then T is a hypercyclic n-tuple.
A strictly increasing sequence of positive integers {nk}k is said to be syndetic
sequence, if Supk(nk+1 − nk) < ∞.
A tuple T = (T1, T2, ..., Tn) on a space X is called syndetically hypercyclic if
for any syndetic sequences of positive integers
{mk,1}k, {mk,2}k, ..., {mk,n}k
the sequence
{T
mk,1
1 T
mk,2
2 ...Tmk,n
n }k

3. On syndetically hypercyclic Ttuples 2599
is hypercyclic, in other hand, there is x ∈ X such that {Tnk x : k ≥ 0} is dens
in X, that is,
{T
mk,1
1 T
mk,2
2 ...T
mk,n
n (x)} = X.
Given continuous linear operators T1,T2,...,Tn that is T1, T2, ..., Tn ∈ L(X),
defined on a separable F-space X, also suppose that T = (T1, T2, ..., Tn) be
a tuple of operators T1,T2,...,Tn, Then T satisfies the Hypercyclicity Crite-
rion if and only if for any strictly increasing sequences of positive integers
{mk,1}k, {nk,2}k, ..., {mk,n}k such that
Supk(mk+1,j − mk,j) < ∞
for all j, then the sequence {T
mk,1
1 T
mk,2
2 ...T
mk,n
n }k is hypercyclic. Also, for each
hypercyclic vector x ∈ X of T, there exists two strictly increasing sequence
{mk}k,{nk}k such that
Supk(mk+1,j − nk,j) < ∞
for all j, and {T
mk,1
1 T
mk,2
2 ...T
mk,n
n }k is somewhere dense, but not dense in X,
That is, the tuple T = (T1, T2, ..., Tn) and the sequence {T
mk,1
1 T
mk,2
2 ...T
mk,n
n }k
do not share the same hypercyclic vectors. Let F be a Frechet space and
T1, T2, ..., Tn are bounded linear operators on F, and T = (T1, T2, ..., Tn) be a
tuple of operators T1, T2, ..., Tn. The space F is called topologically mixing if
for any given open sets U and V , there exist positive numbers M1, M2,...,Mn
such that
T
mi,1
1 T
mi,2
2 ...Tmi,n
n (U) ∩ V = φ , ∀mi,j ≥ Mj , i = 1, 2, ..., n
Notice that, If the tuple T satisfies the hypercyclic criterion for syndetic se-
quences, then T is topologically mixing tuple on space F. Let V be a topolog-
ical vector space(TV S) and T1, T2, ..., Tn are bounded linear operators on V ,
and T = (T1, T2, ..., Tn) be a tuple of operators T1, T2, ..., Tn. The tuple T is
called weakly mixing if
T × T × ... × T : X × X × ... × X → X × X × ... × X
is topologically transitive.
Theorem 2.2 Let X be a topological vector space(TV S) and T1, T2, ..., Tn are
continuous mapping on X, and T = (T1, T2, ..., Tn) be a tuple of operators
T1, T2, ..., Tn. Then the following are equivalent:
(i). T is weakly mixing.
(ii). For any pair of non-empty open subsets U,V in X, and for any syndetic
sequences {mk,1}, {mk,2}, ..., {mk,n}, there exist mk,1, mk,2, ..., mk,n such that
T
mk,1
1 T
mk,2
2 ...T
mk,n
n (U) ∩ (V ) = ∅

4. 2600 M. Habibi
(iii). It suffices in (ii) to consider only those sequences {mk,1}, {mk,2}, ..., {mk,n}
for which there is some m1 ≥ 1, m2 ≥ 1,...,mn ≥ 1 with
mk,j ∈ {mj, 2mj}
for all k and for all j.
proof (i) → (ii). Given {mk,1}, {mk,2}, ..., {mk,n} and U,V satisfying the
hypothesis of condition (ii), take
mj = Supk{mk+1,j − mk,j}
for all j and the n−product map
n−tims
T × T... × T :
n−tims
X × X... × X →
n−tims
X × X... × X
is transitive, Then there is mk ,1, mk ,2, ..., mk ,n in N such that
(T
mk ,1
1 T
mk ,2
2 ...T
mk ,n
n (U)) ((T
mk ,1
1 )−1
(T
mk ,2
2 )−1
...(T
mk ,n
n )−1
(V )) = ∅
∀mk ,1 = 1, 2, ..., m, ∀mk ,2 = 1, 2, ..., m, ..., ∀mk ,n = 1, 2, ..., m
so
(T
mk ,1+mk ,1
1 T
mk ,2+mk ,2
2 ...T
mk ,n+mk ,n
n (U)) (V )) = ∅
∀mk ,1 = 1, 2, ..., m, ∀mk ,2 = 1, 2, ..., m, ..., ∀mk ,n = 1, 2, ..., m
By the assumption on {mk,1}, {mk,2}, ..., {mk,n},for all j, we have
{mk,j : k ∈ N} ∩ {n + 1, n + 2, ..., n + mj} = ∅
If for all j we select mk,j ∈ {mk,j : k ∈ N} ∩ {n + 1, n + 2, ..., n + mj} then
we have T
mk,1
1 T
mk,2
2 ...T
mk,n
n (U) ∩ (V ) = ∅, by this the proof of (i) → (ii) is
completed.
The case (ii) → (iii) is trivial.
Case (iii) → (i). Suppose that U, V1, V2 are non-empty open subsets of X,
then there are {mk,1}, {mk,2}, ..., {mk,n} in N such that
T
mk,1
1 T
mk,2
2 ...Tmk,n
n U ∩ V1 = ∅
T
mk,1
1 T
mk,2
2 ...Tmk,n
n U ∩ V2 = ∅.
This will imply that T is weakly mixing. Since (iii) is satisfied, then we can
take {mk,1}, {mk,2}, ..., {mk,n} in N such that
T
mk,1
1 T
mk,2
2 ...Tmk,n
n V1 ∩ V2 = ∅

5. On syndetically hypercyclic Ttuples 2601
By continuity, we can find V1 ⊂ V1 open and non-empty such that
T
mk,1
1 T
mk,2
2 ...Tmk,n
n V1 ⊂ V2.
Also there exist some mk ,1, mk ,2, ..., mk ,n in N such that
T
mk ,1+η1
1 T
mk ,2+η2
2 ...T
mk ,n+ηn
n U ⊂ V1
for ηj = 0, mj we take mk,j = mk ,j + ηj, for all j, indeed we find strictly
increasing sequences of positive integers mk,1, mk,2, ..., mk,n such that
mk,j ∈ {mj, 2mj}
for all j,and
T
mk,1
1 T
mk,2
2 ...Tmk,n
n U ∩ V1 = ∅, ∀k ∈ N
Now we have
T
mk ,1+η1
1 T
mk ,2+η2
2 ...T
mk ,n+ηn
n U V1 = ∅
So the set
∅ = T
mk,1
1 T
mk,2
2 ...Tmk,n
n (T
mk,1
1 T
mk,2
2 ...T
mk,n
n U ∩ V1)
is a subset of
(T
mk ,1+η1
1 T
mk ,2+η2
2 ...T
mk ,n+ηn
n U) ∩ (T
mk,1
1 T
mk,2
2 ...Tmk,n
n V1)
then we have
T
mk,1
1 T
mk,2
2 ...Tmk,n
n (U) (V1) = φ
and similarly
T
mk,1
1 T
mk,2
2 ...Tmk,n
n (U) (V2) = φ
now, this is the end of proof.
References
[1] J. Bes and A. Peris, Hereditarily hypercylic operators, Jour. Func. Anal.
, 1 (167) (1999), 94-112.
[2] M. Habibi, n-Tuples and chaoticity, Int. Journal of Math. Analysis, 6 (14)
(2012), 651-657 .
[3] M. Habibi, ∞-Tuples of Bounded Linear Operators on Banach Space, Int.
Math. Forum, 7 (18) (2012), 861-866.
[4] M. Habibi and F. Safari, n-Tuples and Epsilon Hypercyclicity, Far East
Jour. of Math. Sci. , 47 (2) (2010), 219-223.

6. 2602 M. Habibi
[5] M. Habibi and B. Yousefi, Conditions for a tuple of operators to be topo-
logically mixing, Int. Jour. of App. Math. , 23(6) (2010), 973-976.
[6] A. Peris and L. Saldivia, Syndentically hypercyclic operators, Integral
Equations Operator Theory , 51, No. 2 (2005) 275-281.
[7] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. , 347
(1995), 993-1004.
[8] B. Yousefi and M. Habibi, Syndetically Hypercyclic Pairs, Int. Math. Fo-
rum , 5 (66) (2010), 3267 - 3272.
[9] B. Yousefi and M. Habibi, Hereditarily Hypercyclic Pairs, Int. Jour. of
App. Math. , 24(2) (2011)), 245-249.
[10] B. Yousefi and M. Habibi, Hypercyclicity Criterion for a Pair of Weighted
Composition Operators, Int. Jour. of App. Math. , 24 (2) (2011), 215-
219.
Received: May, 2012