Weighted Composition Operators and Supercyclicity Criterion

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Volume 2011, Article ID 514370,5pages doi:10.1155/2011/514370

Research Article

Weighted Composition Operators and

Supercyclicity Criterion

Bahmann Yousefi

1

_{and Javad Izadi}

2

1_{Department of Mathematics, Payame Noor University, P.O. Box: 71955-1368 Shiraz, Iran} 2_{Department of Mathematics, Payame Noor University, P.O. Box: 19395-4697 Tehran, Iran}

Correspondence should be addressed to Bahmann Yousefi,b yousefi@pnu.ac.ir

Received 16 April 2011; Revised 6 June 2011; Accepted 15 June 2011

Academic Editor: Naseer Shahzad

Copyrightq2011 B. Yousefi and J. Izadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider an equivalent condition to the property of Supercyclicity Criterion, and then we investigate this property for the adjoint of weighted composition operators acting on Hilbert spaces of analytic functions.

1. Introduction

LetTbe a bounded linear operator onH. Forx∈H, the orbit ofxunderTis the set of images ofxunder the successive iterates ofT:

orbT, x x, Tx, T2_{x, . . .}_. _1.1

The vectorxis called supercyclic forTifCorbT, xis dense inH. Also a supercyclic operator is one that has a supercyclic vector. For some sources on these topics, see1–16.

LetH be a separable Hilbert space of functions analytic on a plane domainG such that, for eachλinG, the linear functional of evaluation atλgiven byf → fλis a bounded linear functional onH. By the Riesz representation theorem, there is a vectorKλinHsuch

thatfλ f, Kλ. We callKλthe reproducing kernel atλ.

A complex-valued functionϕonGis called a multiplier ofHifϕH⊂H. The operator of multiplication byϕis denoted byMϕand is given byf → ϕf.

Ifϕis a multiplier ofHandψis a mapping fromGintoG, thenCϕ,ψ:H → Hby

Cϕ,ψfz ϕzfψz 1.2

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The holomorphic self-maps of the open unit diskDare divided into classes of elliptic and nonelliptic. The elliptic type is an automorphism and has a fixed point inD. It is well known that this map is conjugate to a rotationz → λzfor some complex numberλwith |λ|1. The maps of those which are not elliptic are called of non-elliptic type. The iterate of a non-elliptic map can be characterized by the Denjoy-WolﬀIteration theorem.

2. Main Results

We will investigate the property of Hypercyclicity Criterion for a linear operator and in the special case, we will give suﬃcient conditions for the adjoint of a weighted composition operator associated with elliptic composition function which satisfies the Supercyclicity Criterion.

Theorem 2.1Supercyclicity Criterion. LetHbe a separable Hilbert space andT is a continuous

linear mapping onH. Suppose that there exist two dense subsetsY andZinH, a sequence{nk}of

positive integers, and also there exist mappingsSnk :Z → Hsuch that

1TnkSnkz → zfor everyz∈Z,

2||Tnky||||Snkz|| → 0 for everyy∈Y and everyz∈Z.

Then,T is supercyclic.

If an operatorTholds in the assumptions ofTheorem 2.1, then one says thatTsatisfies the Supercyclicity Criterion.

Definition 2.2. Let T be a bounded linear operator on a Hilbert space H. We refer to

n≥1KerTnas the generalized kernel ofT.

Theorem 2.3. Let T be a bounded linear operator on a separable Hilbert space H with dense

generalized kernel. Then, the following conditions are equivalent:

1T has a dense range,

2T is supercyclic,

3T satisfies the Supercyclicity Criterion.

Proof. See2, Corollary 3.3.

Remark 2.4. In2, for the proof of implication1 → 3ofTheorem 2.3, it has been shown

that TT is supercyclic which implies by using Lemma 3.1 in 2 that T satisfies the Supercyclicity Criterion. This implication can be proved directly without using Lemma 3.1 in2, as follows: IfTis a bounded linear operator on a separable Hilbert spaceHwith dense range and dense generalized kernel, then it follows thatTis supercyclic1, Exercise 1.3. Now suppose thath0is a supercyclic vector ofT. SetX0 CorbT, h0andY0 the generalized

kernel ofT. SinceT is supercyclic, there exist sequences{nj}_j ⊂N,{αj}_j ⊂Cand{fj}_j ⊂H

such thatfj → 0 andαjTnjfj → h0. DefineSnk :X0 → Hby

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Then, clearly,TnkSnk → IX0pointwise onX0and

Tnk_y_S

nkx −→0 2.2

for everyy∈Y0and everyx∈X0. Hence,Tsatisfies the Supercyclicity Criterion.

From now on letH be a Hilbert space of analytic functions on the open unit discD such thatHcontains constants and the functional of evaluation atλis bounded for allλin

D. Also letϕ :D → Cbe a nonconstant multiplier ofHand letψbe an analytic map from

D intoD such that the composition operatorCψ is bounded on H. We define the iterates

ψn ψ◦ψ◦ · · · ◦ψ ntimes. Byψ−n1orψ−nwe mean thenth iterate ofψ−1, henceψnm ψmn

form−1,1.

Definition 2.5. We say that{zn}n≥0is a B-sequence forψifψzk zk−1for allk≥1.

Corollary 2.6. Suppose that{zn}n≥0⊂Dis a B -sequence forψand has limit point inD. Ifϕz0 0,

thenC∗_ϕ,ψsatisfies the Supercyclicity Criterion.

Proof. PutACϕ,ψ. Sinceϕz0 0, we getKzi ∈KerA∗nfor alli0, ..., n−1. HenceA∗has

dense generalized kernel. Now letf, A∗Kzn 0 for alln, thusϕzn·f◦ψzn 0 for all

n. This implies thatfis the zero constant function, becauseϕis nonconstant and{zn}n≥0has

limit point inD. Thus,A∗has dense range and, byTheorem 2.3, the proof is complete.

Example 2.7. Letψz e−√2πiz,ϕz z−1/2, and definezn 1/2e

√

2nπi _{for all}_n_≥_0.

Now byCorollary 2.6, the operatorC_ϕ,ψ∗ satisfies the Supercyclicity Criterion.

Theorem 2.8. Letψbe an elliptic automorphism with interior fixed pointpandϕ:D → Csatisfies

the inequality|ϕp|<1≤ |ϕz|for allzin a neighborhood of the unit circle. Then, the operatorC∗_ϕ,ψ

satisfies the Supercyclicity Criterion.

Proof. PutΨ αp◦ψ◦αpandΦ ϕ◦αpwhere

αpz p−z

1−pz. 2.3

SinceΨis an automorphism withΨ0 0, thusΨis a rotationz → eiθzfor someθ∈0,2π and everyz ∈U. SetT C∗_Φ,ΨandS C_ϕ,ψ∗ . Then, clearlyS∗ CαpT∗C−αp1, thusT is similar

toSwhich implies thatS satisfies the Supercyclicity Criterion if and only ifT satisfies the Supercyclicity Criterion. Since|αpz| → 1− when|z| → 1−, so|Φ0| < 1 ≤ |Φz|for all

zin a neighborhood of the unit circle. So, without loss of generality, we suppose that ψ is a rotationz → eiθzand |ϕ0| < 1 ≤ |ϕz|for allz in a neighborhood of the unit circle. Therefore, there exist a constantλand a positive numberδ <1 such that|ϕz|< λ <1 when |z|< δ, and|ϕz| ≥1 when|z|>1−δ. SetU1{z:|z|< δ}andU2{z:|z|>1−δ}. Also,

consider the sets

H1span{Kz:z∈U1},

H2span{Kz:z∈U2},

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where span {·} is the set of finite linear combinations of {·}. By using the Hahn-Banach theorem,H1 and H2 are dense subsets ofH. Sinceψ is a rotation, the sequence{ψnmλ}n

is a subset of the compact set{z :|z| λ}for eachλinDandm−1,1. Now by, using the Banach-Steinhaus theorem, the sequence{Kψm

nλ}_nis bounded for eachλinDandm−1,1.

Note that, for eachZ,|z||ψnz|. So, ifz ∈U1, then|ϕψiz|< λ < 1 and ifz∈ U2, then

|ϕψ−1

i z| ≥1 for each positive integeri. Also, note that

Sn_K_z n−1 i0

ϕψiz

K_ψnz 2.5

for every positive integernandz∈Dsee12. Now, ifz∈U1, thenSnKz → 0 asn → ∞.

Therefore the sequence{Sn}converges pointwise to zero on the dense subsetH1. Define a

sequence of linear mapsWn:H2 → H2by extending the definition

WnKz n

i1

ϕψ−1

i z

−1

K_ψ_n−1_z 2.6

z ∈ U2linearly toH2. Note that, for allz ∈ U2, the sequence {WnKz}n is bounded and

Sn_W

nKzKzonH2which implies thatSnWnis identity on the dense subsetH2. Hence,

Sn_f_W

ng−→0 2.7

for everyf ∈H1and everyg∈H2. Now, byTheorem 2.1, the proof is complete.

Corollary 2.9. Under the conditions ofTheorem 2.8,C∗_ϕ,ψC_ϕ,ψ∗ is supercyclic.

Proof. It is clear sinceC∗_ϕ,ψsatisfies the Supercyclicity Criterion.

Example 2.10. Let ϕz 3/2z and ψz eiθz. Then, the operator C_ϕ,ψ∗ satisfies the

Supercyclicity Criterion, because 0 is an interior fixed point ofψ, andϕ0 < 1 ≤ |ϕz|for |z|>2/3.

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