2. KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI204
( ) ., XT =xOrb
The tuple ( )nTTT ...,,, 21=T is called topologically mixing if for any given open
subsets U and V of ,X there exist positive numbers nMMM ...,,, 21 such that
( ) ii
m
n
mm
MmTTT n ≥∀∅≠ ,21
21 VU ∩ for ....,,2,1 ni =
For easy in this paper, we take 2=n in tuples and by the pairs, we mean
2-tuples. By this, the pair ( )21, TT=T is called topologically mixing if for any
given open subsets U and V of ,X there exist two positive numbers N and M such
that
( ) .,,21 NnMmTT nm
≥∀≥∀∅≠VU ∩ (1)
A nice criterion, namely, the Hypercyclicity Criterion is used in the proof of our
main theorem. It was developed independently by Kitai, Gethner and Shapiro. This
criterion has used to show that hypercyclic operators arise within the class of
composition operators, weighted shifts, adjoints of multiplication operators, and
adjoints of subnormal and hyponormal operators, and Hereditarily operators,
topologically mixing. The formulation of the Hypercyclicity Criterion in the
following theorem was given by J. Bes in Ph.D. thesis. Readers can see [1-11] for
some more information. Note that, all of the operators in this paper are commutative
bounded linear operators on a Fréchet space.
2. Main Result
Theorem 2.1 (The Hypercyclicity Criterion). Let X be a separable Banach
space and ( )nTTTT ...,,, 21= be an n-tuple of continuous linear mappings on X. If
there exist two dense subsets Y and Z in X, and strictly increasing sequences
{ } { } { }∞
=
∞
=
∞
= 1,12,11, ...,,, jnjjjjj mmm such that:
1. 0,2,1,
21 →njjj m
n
mm
TTT on Y as ,∞→j
2. There exist functions { }XZS j →: such that for every ,Zz ∈ ,0→zS j
and ,,2,1,
21 zzSTTT j
m
n
mm njjj
→
then T is a hypercyclic n-tuple.
3. ON n-TUPLES OF WEIGHTED SHIFTS … 205
Theorem 2.2. Let nTTT ...,,, 21 be hypercyclic operators on a Fréchet space
,F and assume that ( )( )nTTTT ...,,, 21==T be a hypercyclic tuple of =T
( )....,,, 21 nTTT If the tuple T satisfies the hypercyclic criterion for a syndetic
sequence, then T is topologically mixing tuple.
Theorem 2.3. Let nTTT ...,,, 21 be unilateral weighted backward shifts with
weighted sequences { } { } { }0:...,,0:,0: ,2,1,
≥≥≥ iaiaia niii mmm and suppose
that ( )nTTT ...,,, 21=T be an n-tuple of operators ....,,, 21 nTTT Then T is
topologically mixing if and only if
....,,2,1,lim
1
,
na
k
i
m
k i
=λ∞=∏=
∞→ λ
(2)
Proof. For easy, we take 2=n in our proof and the case ,2>n the proof is
similar. Now, let 1T and 2T be unilateral weighted backward shifts with weighted
sequences { }0: ≥ia in and { }0: ≥ib im and suppose that ( )21, TT=T be a pair
of operators 1T and .2T We deal first with unilateral backward shifts. We show that
if (2) is satisfied, then the pair of unilateral backward weighted shift is topologically
mixing. Indeed, take the following dense set in :2
{{ } }.eventually0:2
=∈= nn xxD
The hypercyclicity criterion applies for DDD == 21 and the maps ,n
n SS =
where 2
: →DS is defined by
( ) ....,,,0...,,
2
2
1
1
21 ⎟
⎠
⎞
⎜
⎝
⎛=
a
x
a
x
xxS
Notice that, the map S may not be well defined either as a map or as a bounded
operator with domain 2
if the sequence { }ia is not bounded away from zero,
however, it always makes sense when we restrict S to the set D. Hence, Theorem
2.2 applies and T is topologically mixing. On the other hand, let us prove that if T
is topologically mixing, then (2) holds. Arguing by contradiction, assume that this is
not true, that is,
4. KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI206
.inflim
1 1
∏∏= =
∞<
k tn
i
m
j
ija
In other words, there exists 0>M such that
.and,
1 1
∏∏= =
∀∀<
k tn
i
m
j
ij tkMa
Consider ( ) 2
1 ...,0,0,1 ∈=e (note that ( ),...,0,1,0...,,0=ie so that the
element 1 is ith component). Let
2
1
<ε and take .
2
1
M
<δ Let U be the ball of
radius δ and centered at the origin and let V be the ball of radius ε centered at .1e
Since we are assuming that T is topologically mixing, (1) is satisfied. Take
Mnk > and .Nmj > Thus ( ) ,21 ∅≠VU ∩jk mn
TT for all Mnk > and ,Nmj >
therefore there is a vector { } U∈= nxx such that { }( ) .21 V∈n
mn
xTT jk
Let knx and jmx be the kn -component and jm -component of x. It follows that
1δ<knx and .2δ<jmx On the other hand,
( )
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
= ∏∏= =
k j
kj
jk
n
i
n
j
ni
mn
xaxTT
2 1
21 ...,
and notice that
.
2
1
2 1
21∏∏= =
<δδ<
k j
kj
n
i
n
j
ni Mxa
In particular,
( ) ,
2
1
1
2 1
121 ε>>−≥− ∏∏= =
k j
kj
jk
n
i
n
j
ni
mn
xaexTT
a contradiction. Now, we treat the case of bilateral weighted backward shifts. If (4)
and (5) hold, consider the dense set in ( ):2 Z
{{ } }.someforif0:2
kknxxD nn >=∈=
5. ON n-TUPLES OF WEIGHTED SHIFTS … 207
As before, the Hypercyclicity Criterion applies for DDD == 21 and the maps
,n
n SS = where ( )Z2
: →DS is defined by ( ) 1
1
+= i
i
e e
a
x
xS i
and the sequences
{ } Nnk = and { } .Nnj = Therefore, Theorem 2.2 applies and T is topologically
mixing. Let us prove that if T is topologically mixing, then (4) and (5) hold.
We will argue by a contradiction. The case ( )∏ =
∞<
n
i in a
1
inflim leads to a
contradiction as we did for the unilateral shift. Therefore, assume that
( )∏ =
>
n
i in a
1
.0suplim Hence, there exist ,0>c sequences ∞→kn and ∞→jn
such that
( )( ) .0
0 0
11∏∏= =
−− >>
k jn
i
n
j
ji ca
Take c<ε<0 and choose 1δ and 2δ so that ( ) ε>δ− 11c and ( ) .1 1 ε>δ−c
Let 1V be the ball of radius 1δ and 2V be the ball of radius 2δ centered at the origin
and let 1U be the ball of radius 1δ centered at 1e and 2U be the ball of radius 2δ
centered at .2e Since T is topologically mixing, there exist 1m and 2m such that
( ) ∅≠VUTT nn
∩21
21 for all 11 mn ≥ and .22 mn ≥ Pick kk mn ≥ and jj mn ≥
and let U∈nx be such that ( ) .
1
2
1
1 V∈
++
n
nn
xTT jk However,
( )n
nn
xTT jk 1
2
1
1
++
>ε
( )≺ k
jk
nn
nn
exTT −
++
≥ ,
1
2
1
1
( )( ) ( ) ,01
0 0
11 1∏ ∏= =
−− >δ−>=
k jn
i
n
j
xji ca
a contradiction. Furthermore, from the proof, we get that a backward shift is
topologically mixing if and only if it satisfies the Hypercyclicity Criterion for a
syndetic sequence. Thus the proof is completed.
Corollary 2.4. Similarly, suppose that ,1T 2T are two bilateral backward shifts
with weighted sequences { }Ziai ∈: and { }Zja j ∈: and suppose ( )21, TTT = is
a pair of operators ,1T .2T Then T is topologically mixing if and only if
6. KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI208
,0lim,lim
1 0
∏ ∏= =
−
∞→∞→
=∞=
n
i
n
i
i
n
i
n
aa (3)
.0lim,lim
1 0
∏ ∏= =
−
∞→∞→
=∞=
n
i
n
i
i
n
i
n
bb (4)
Acknowledgement
This research is partially supported by a grant from Research Council of Islamic
Azad University, branch of Dehdasht. The authors gratefully acknowledge this
support.
References
[1] J. Bes, Three problem on hypercyclic operators, Ph.D. Thesis, Kent State University,
1998.
[2] P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44 (1997), 345-353.
[3] P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators,
Memoirs of Amer. Math. Soc. 125, Amer. Math. Providence, RI, 1997.
[4] G. Costakis and M. Sambarino, Topologically mixing hypercyclic operators, Proc.
Amer. Math. Soc. 132(2) (2004), 385-389.
[5] F. Ershad, B. Yousefi and M. Habibi, Conditions for reflexivity on some sequences
spaces, Int. J. Math. Anal. 4(30) (2010), 1465-1468.
[6] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of
holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288.
[7] M. Habibi and F. Safari, n-tuples and epsilon hypercyclicity, Far East J. Math. Sci.
(FJMS) 47(2) (2010), 219-223.
[8] M. Habibi and B. Yousefi, Conditions for a tuple of operators to be topologically
mixing, Int. J. Appl. Math. 23(6) (2010), 973-976.
[9] C. Kitai, Invariant closed sets for linear operators, Thesis, University of Toronto,
1982.
[10] A. L. Shields, Weighted shift operators and analytic function theory, Math. Surveys,
No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49-128.
[11] B. Yousefi and M. Habibi, Syndetically hypercyclic pairs, Int. Math. Forum 5(66)
(2010), 3267-3272.
![KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI204
( ) ., XT =xOrb
The tuple ( )nTTT ...,,, 21=T is called topologically mixing if for any given open
subsets U and V of ,X there exist positive numbers nMMM ...,,, 21 such that
( ) ii
m
n
mm
MmTTT n ≥∀∅≠ ,21
21 VU ∩ for ....,,2,1 ni =
For easy in this paper, we take 2=n in tuples and by the pairs, we mean
2-tuples. By this, the pair ( )21, TT=T is called topologically mixing if for any
given open subsets U and V of ,X there exist two positive numbers N and M such
that
( ) .,,21 NnMmTT nm
≥∀≥∀∅≠VU ∩ (1)
A nice criterion, namely, the Hypercyclicity Criterion is used in the proof of our
main theorem. It was developed independently by Kitai, Gethner and Shapiro. This
criterion has used to show that hypercyclic operators arise within the class of
composition operators, weighted shifts, adjoints of multiplication operators, and
adjoints of subnormal and hyponormal operators, and Hereditarily operators,
topologically mixing. The formulation of the Hypercyclicity Criterion in the
following theorem was given by J. Bes in Ph.D. thesis. Readers can see [1-11] for
some more information. Note that, all of the operators in this paper are commutative
bounded linear operators on a Fréchet space.
2. Main Result
Theorem 2.1 (The Hypercyclicity Criterion). Let X be a separable Banach
space and ( )nTTTT ...,,, 21= be an n-tuple of continuous linear mappings on X. If
there exist two dense subsets Y and Z in X, and strictly increasing sequences
{ } { } { }∞
=
∞
=
∞
= 1,12,11, ...,,, jnjjjjj mmm such that:
1. 0,2,1,
21 →njjj m
n
mm
TTT on Y as ,∞→j
2. There exist functions { }XZS j →: such that for every ,Zz ∈ ,0→zS j
and ,,2,1,
21 zzSTTT j
m
n
mm njjj
→
then T is a hypercyclic n-tuple.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)