paperno11-ghezelbashhabibisafari-ijmms

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: [email protected]



Branch of Dehdasht

P. O. Box 7164754818, Shiraz, Iran




Branch of Dehdasht

P. O. Box 7571734494, Dehdasht, Iran


Abstract

In this paper, we characterize the topologically mixing to a pair of

unilateral backward weighted shifts on a Fréchet space.


50

1. Introduction

Let X be a Fréchet space and ( )nTTTT ...,,, 21= be an n-tuple of operators.

Then we will let

{ }0:...,,, 2121 ≥= i

kn

kkkTTT 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,

( ) ., X�T =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 ∩nmTT ,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 nnTT

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 jnn

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: ≥iain 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

nk i

a

1

,lim ∏=

∞→∞=

k

i

mk 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

in

a

1

,lim .0lim

0

∏=

−∞→

=

n

i

in

a (3)

∏=

∞→∞=

n

i

in

b

1

,lim ∏=

−∞→

=

n

i

in

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 ,nn SS =

where 2: �→DS is defined by

( ) ....,,,0...,,2

2

1

121

=

a

x

a

xxxS


52

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 ( ) 21 ...,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

mnTT for all Mnk > and ,Nm j >

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

mnxTT

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

mnxaxTT

2 1

21 ...,

and notice that,

∏∏= =

<δδ<k j

jk

n

i

n

j

ni Mxa

2 1

21 .2

1

ON TOPOLOGICALLY MIXING OF n-TUPLS

53

In particular,

( ) ε>>−≥− ∏∏= =

2

11

2 1

121

k j

jk

jk

n

i

n

j

ni

mnxaexTT

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

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

1+= i

ie e

a

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) are hold. We

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

n

i in a1

inflim leads to a

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

( )∏ =>

n

i in a1

.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

( ) ,2121 φ≠VUTTnn

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

and let U∈nx be so that ( ) .1

21

1 V∈++

n

nnxTT

jk However,

( )n

nnxTT

jk1

21

1

++>ε

( ) �≺k

jknn

nnexTT −

++≥ ,

1

21

1


54

( ) ( ) ( )∏ ∏= =−− >δ−>=

k jn

i

n

jxji ca

0 011 01

1

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. ~

References

[1] J. Bes, Three problems on hypercyclic operators, Ph.D. thesis, Kent State University,

1998.

[2] J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167(1) (1999),

94-112.

[3] P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44(2) (1997),

345-353.

[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 sequence

spaces, Int. J. Math. Anal. (Ruse) 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(2) (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, Topics in

Operator Theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974,

pp. 49-128.

[11] B. Yousefi, Bounded analytic structure of the Banach space of formal power series,

Rend. Circ. Mat. Palermo (2) 51(3) (2002), 403-410.

[12] B. Yousefi and M. Habibi, Syndetically hypercyclic pairs, International Mathematical

Forum 5(66) (2010), 3267-3272.

[13] B. Yousefi and M. Habibi, Hereditarily hypercyclic pairs, Int. J. Appl. Math. 24(2)

(2011), 245-249.

[14] B. Yousefi and M. Habibi, Hypercyclicity criterion for a pair of weighted composition

operators, Int. J. Appl. Math. 24(2) (2011), 215-219.

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