1. Pioneer Journal of Mathematics and Mathematical Sciences
Volume 3, Number 1, 2011, Pages 49-54
This paper is available online at http://www.pspchv.com/content_PJMMS.html
;20119,DecemberReceived Revised July 10, 2011
2010 Mathematics Subject Classification: 37A25, 47B37.
Keywords and phrases: hypercyclic vector, hypercyclicity criterion, topologically mixing, pair
of shifts, backward shift, weighted shift.
This research is partially supported by a grant from Research Council of Meymeh Branch,
Islamic Azad University, so the authors gratefully acknowledge this support.
© 2011 Pioneer Scientific Publisher
ON TOPOLOGICALLY MIXING
OF n-TUPLS
FATEMEH GHEZELBASH, MEZBAN HABIBI and FATEMEH SAFARI
Department of Mathematics
Islamic Azad University
Meymeh Branch, Meymeh
P. O. Box 8196983443, Isfahan, Iran
e-mail: ghezelbash@iaumeymeh.ac.ir
Department of Mathematics
Islamic Azad University
Branch of Dehdasht
P. O. Box 7164754818, Shiraz, Iran
e-mail: habibi.m@iaudehdasht.ac.ir
Department of Mathematics
Islamic Azad University
Branch of Dehdasht
P. O. Box 7571734494, Dehdasht, Iran
e-mail: safari.s@iaudehdasht.ac.ir
Abstract
In this paper, we characterize the topologically mixing to a pair of
unilateral backward weighted shifts on a Fréchet space.

2. FATEMEH GHEZELBASH, MEZBAN HABIBI and FATEMEH SAFARI50
1. Introduction
Let X be a Fréchet space and ( )nTTTT ...,,, 21= be an n-tuple of operators.
Then we will let
{ }0:...,,, 21
21 ≥= i
k
n
kk
kTTT nF
be the semigroup generated by T. For X∈x we take
( ) { }.:, FT ∈= SSxxOrb
The set ( )xOrb ,T is called orbit of vector x under T and Tuple =T
( )nTTT ...,,, 21 is called hypercyclic pair if the set ( )xOrb ,T is dense in ,X that is,
( ) ., XT =xOrb
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 N and M such that
( ) ,21 φ≠VU ∩nm
TT ,Mm ≥∀ .Nn ≥∀ (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 Ph.D thesis. Readers can see [1-14] for some
information.
2. Main Result
Theorem 2.1 (The hypercyclicity criterion). Suppose X is a separable Banach
space and ( )21, TT=T is a pair of continuous linear mapping on .X If there exist
two dense subsets Y and Z in X and two strictly increasing sequences { }jn and
{ }kn such that
1. .021 →kj nn
TT

3. ON TOPOLOGICALLY MIXING OF n-TUPLS 51
2. There exist functions { }XZ →:jS such that for every ,Z∈z ,0→zS j
and ,21 zzSTT j
nn kj
→ then T is hypercyclic pair.
Theorem 2.2. Let ,1T 2T be two hypercyclic operators on a frechet space ,F
and assume that ( )21, TT=T be a hypercyclic pair of 1T and .2T If the pair T
satisfies the hypercyclic criterion for a syndetic sequence, then T is topologically
mixing pair.
Theorem 2.3. 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
is a pair of operators 1T and .2T Then T is topologically mixing if and only if
∏=
∞→
∞=
k
i
n
k i
a
1
,lim ∏=
∞→
∞=
k
i
m
k i
b
0
.lim (2)
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
∏=
∞→
∞=
n
i
i
n
a
1
,lim .0lim
0
∏=
−
∞→
=
n
i
i
n
a (3)
∏=
∞→
∞=
n
i
i
n
b
1
,lim ∏=
−
∞→
=
n
i
i
n
b
0
.0lim (4)
Proof. 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

4. FATEMEH GHEZELBASH, MEZBAN HABIBI and FATEMEH SAFARI52
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,
∏∏= =
∞<
k tn
i
m
j
ija
1 1
.inflim
In other words, there exist 0>M such that,
∏∏= =
∀<
k tn
i
m
j
ij kMa
1 1
, and .t∀
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, then (1) is satisfy. Take
Mnk > and .Nm j > Thus ( ) ,21 φ≠VU ∩jk mn
TT for all Mnk > and ,Nm j >
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 In the other hand,
( )










= ∏∏= =
k j
jk
jk
n
i
n
j
ni
mn
xaxTT
2 1
21 ...,
and notice that,
∏∏= =
<δδ<
k j
jk
n
i
n
j
ni Mxa
2 1
21 .
2
1

5. ON TOPOLOGICALLY MIXING OF n-TUPLS 53
In particular,
( ) ε>>−≥− ∏∏= =
2
1
1
2 1
121
k j
jk
jk
n
i
n
j
ni
mn
xaexTT
is a contradiction. Now, we treat the case of bilateral weighted backward shifts. If (4)
and (5) are holds, consider the dense set in ( ):2 Z
{{ } }.someforif0:2
kknxxD nn >=∈=
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 { } .Nn j = Therefore, Theorem 2.2 applies and T is topologically
mixing. Let us prove that if T is topologically mixing, then (4) and (5) are hold. We
will argue by 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 and sequences ∞→kn and
∞→jn such that
( )( )∏∏= =
−− >>
k jn
i
n
j
ji ca
0 0
11 .0
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 1U be the ball of radius 2δ
centered at .2e Since T is topologically mixing, there exists 1m and 2m such that
( ) ,21
21 φ≠VUTT
nn
∩ for all 11 mn ≥ and .22 mn ≥ Pick kk mn ≥ and jj mn ≥
and let U∈nx be so 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

6. FATEMEH GHEZELBASH, MEZBAN HABIBI and FATEMEH SAFARI54
( )( ) ( )∏ ∏= =
−− >δ−>=
k jn
i
n
j
xji ca
0 0
11 011
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. In this way the proof is completed. ~
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