Far East Journal of Mathematical Sciences (FJMS) Volume 52, Number 2, 2011, Pages 203-208 Published Online: May 31, 2011 This paper is available online at http://pphmj.com/journals/fjms.htm © 2011 Pushpa Publishing House

:tionClassificaject Sub sMathematic 2010 37A25, 47B37. Keywords and phrases: hypercyclic vector, hypercyclicity criterion, topologically mixing,

tuple of shifts, backward shift, weighted shift.

Received January 18, 2011

ON n-TUPLES OF WEIGHTED SHIFTS TO BE TOPOLOGICALLY MIXING

KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI

Islamic Azad University Branch of Dehdasht P.O. Box 7571763111 Dehdasht, Iran e-mail: ostad.k@iaudehdasht.ac.ir

habibi.m@iaudehdasht.ac.ir safari.s@iaudehdasht.ac.ir

Abstract

In this paper, we characterize the topologically mixing to a tuple of unilateral backward weighted shifts on a Fréchet space.

1. Introduction

Let X be a Fréchet space and ( )nTTT ...,,, 21=T be an n-tuple of commutative

bounded linear operators ....,,, 21 nTTT Let

{ }0:2121 ≥= i

kn

kk kTTT nF

be the semigroup generated by .T For ,X∈x we take ( ) { }.:, FT ∈= SSxxorb

The set ( )xOrb ,T is called the orbit of vector x under T and the tuple =T

( )nTTT ...,,, 21 is called hypercyclic tuple if the set ( )xOrb ,T is dense in ,X that

is,

KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI 204

( ) ., 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

( ) iim

nmm 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

nmm

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

mn

mm njjj →

then T is a hypercyclic n-tuple.

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,lim1

, nak

im

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 ,nn SS =

where 2: →DS is defined by

( ) ....,,,0...,,22

11

21 ⎟⎠⎞

⎜⎝⎛= a

xaxxxS

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,

KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI 206

.inflim1 1∏∏= =

∞<k tn

i

m

jija

In other words, there exists 0>M such that

.and,1 1∏∏= =

∀∀<k tn

i

m

jij tkMa

Consider ( ) 21 ...,0,0,1 ∈=e (note that ( ),...,0,1,0...,,0=ie so that the

element 1 is ith component). Let 21<ε and take .2

1M<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 .Nm j > Thus ( ) ,21 ∅≠VU ∩jk mn TT for all Mnk > and ,Nm j >

therefore there is a vector { } U∈= nxx such that { }( ) .21 V∈nmn 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

kjjk

n

i

n

jni

mn xaxTT2 1

21 ...,

and notice that

.21

2 121∏∏

= =

<δδ<k j

kj

n

i

n

jni Mxa

In particular,

( ) ,211

2 1121 ε>>−≥− ∏∏

= =

k j

kjjk

n

i

n

jni

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 >=∈=

ON n-TUPLES OF WEIGHTED SHIFTS … 207

As before, the Hypercyclicity Criterion applies for DDD == 21 and the maps

,nn SS = where ( )Z2: →DS is defined by ( ) 1

1+= i

ie ea

xxS 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) hold.

We will argue by a contradiction. The case ( )∏ =∞<

ni in a1inflim leads to a

contradiction as we did for the unilateral shift. Therefore, assume that

( )∏ =>

ni in a1 .0suplim Hence, there exist ,0>c sequences ∞→kn and ∞→jn

such that

( ) ( ) .00 0

11∏∏= =

−− >>k jn

i

n

jji 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 ∩2121 for all 11 mn ≥ and .22 mn ≥ Pick kk mn ≥ and jj mn ≥

and let U∈nx be such that ( ) .1

21

1 V∈++

nnn xTT jk However,

( )nnn xTT jk 1

21

1++>ε

( )≺ kjk

nnnn exTT −++≥ ,1

21

1

( ) ( ) ( ) ,010 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

KOBRA OSTAD, MEZBAN HABIBI and FATEMEH SAFARI 208

,0lim,lim1 0∏ ∏= =

−∞→∞→

=∞=n

i

n

ii

ni

naa (3)

.0lim,lim1 0∏ ∏= =

−∞→∞→

=∞=n

i

n

ii

ni

nbb (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

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