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

Let T be a bounded linear operator on H. For x∈H, the orbit of x under T is the set of images of x under the successive iterates of T: orb(T,x)={x,Tx,T2x,…}.
The vector x is called supercyclic for T if ℂorb(T,x) is dense in H. Also a supercyclic operator is one that has a supercyclic vector. For some sources on these topics, see [1–16].

Let H be a separable Hilbert space of functions analytic on a plane domain G such that, for each λ in G, the linear functional of evaluation at λ given by f→f(λ) is a bounded linear functional on H. By the Riesz representation theorem, there is a vector Kλ in H such that f(λ)=〈f,Kλ〉. We call Kλ the reproducing kernel at λ.

A complex-valued function φ on G is called a multiplier of H if φH⊂H. The operator of multiplication by φ is denoted by Mφ and is given by f→φf.

If φ is a multiplier of H and ψ is a mapping from G into G, then Cφ,ψ:H→H by Cφ,ψ(f)(z)=φ(z)f(ψ(z))
for every f∈H and z∈G is called a weighted composition operators.

The holomorphic self-maps of the open unit disk 𝔻 are divided into classes of elliptic and nonelliptic. The elliptic type is an automorphism and has a fixed point in 𝔻. It is well known that this map is conjugate to a rotation z→λz for some complex number λ with |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-Wolff Iteration 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 2.1 (Supercyclicity Criterion).

Let H be a separable Hilbert space and T is a continuous linear mapping on H. Suppose that there exist two dense subsets Y and Z in H, a sequence {nk} of positive integers, and also there exist mappings Snk:Z→H such that

TnkSnkz→z for every z∈Z,

||Tnky||||Snkz||→0 for every y∈Y and every z∈Z.

Then, T is supercyclic.

If an operator T holds in the assumptions of Theorem 2.1, then one says that T satisfies the Supercyclicity Criterion.

Definition 2.2.

Let T be a bounded linear operator on a Hilbert space H. We refer to ⋃n≥1Ker(Tn) as the generalized kernel of T.

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:

T has a dense range,

T is supercyclic,

T satisfies the Supercyclicity Criterion.

Proof.

See [2, Corollary 3.3].

Remark 2.4.

In [2], for the proof of implication (1)→(3) of Theorem 2.3, it has been shown that T⨁T 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 in [2], as follows: If T is a bounded linear operator on a separable Hilbert space H with dense range and dense generalized kernel, then it follows that T is supercyclic [1, Exercise 1.3]. Now suppose that h0 is a supercyclic vector of T. Set X0=ℂorb(T,h0) and Y0= the generalized kernel of T. Since T is supercyclic, there exist sequences {nj}j⊂ℕ, {αj}j⊂ℂ and {fj}j⊂H such that fj→0 and αjTnjfj→h0. Define Snk:X0→H by
Snk(αTmh0)=ααkTmfk.
Then, clearly, TnkSnk→IX0 pointwise on X0 and
‖Tnky‖‖Snkx‖⟶0
for every y∈Y0 and every x∈X0. Hence, T satisfies the Supercyclicity Criterion.

From now on let H be a Hilbert space of analytic functions on the open unit disc 𝔻 such that H contains constants and the functional of evaluation at λ is bounded for all λ in 𝔻. Also let φ:𝔻→C be a nonconstant multiplier of H and let ψ be an analytic map from 𝔻 into 𝔻 such that the composition operator Cψ is bounded on H. We define the iterates ψn=ψ∘ψ∘⋯∘ψ (n times). By ψn-1 or ψ-n we mean the nth iterate of ψ-1, hence ψnm=ψmn for m=-1,1.

Definition 2.5.

We say that {zn}n≥0 is a B-sequence for ψ if ψ(zk)=zk-1 for all k≥1.

Corollary 2.6.

Suppose that {zn}n≥0⊂𝔻 is a B -sequence for ψ and has limit point in 𝔻. If φ(z0)=0, then Cφ,ψ* satisfies the Supercyclicity Criterion.

Proof.

Put A=Cφ,ψ. Since φ(z0)=0, we get Kzi∈Ker(A*)n for all i=0,...,n-1. Hence A* has dense generalized kernel. Now let 〈f,A*Kzn〉=0 for all n, thus φ(zn)·f∘ψ(zn)=0 for all n. This implies that f is the zero constant function, because φ is nonconstant and {zn}n≥0 has limit point in 𝔻. Thus, A* has dense range and, by Theorem 2.3, the proof is complete.

Example 2.7.

Let ψ(z)=e-2πiz, φ(z)=z-(1/2), and define zn=(1/2)e2nπi for all n≥0. Now by Corollary 2.6, the operator Cφ,ψ* satisfies the Supercyclicity Criterion.

Theorem 2.8.

Let ψ be an elliptic automorphism with interior fixed point p and φ:𝔻→C satisfies the inequality |φ(p)|<1≤|φ(z)| for all z in a neighborhood of the unit circle. Then, the operator Cφ,ψ* satisfies the Supercyclicity Criterion.

Proof.

Put Ψ=αp∘ψ∘αp and Φ=φ∘αp where
αp(z)=p-z1-p¯z.
Since Ψ is an automorphism with Ψ(0)=0, thus Ψ is a rotation z→eiθz for some θ∈[0,2π] and every z∈U. Set T=CΦ,Ψ* and S=Cφ,ψ*. Then, clearly S*=CαpT*Cαp-1, thus T is similar to S which implies that S satisfies the Supercyclicity Criterion if and only if T satisfies the Supercyclicity Criterion. Since |αp(z)|→1- when |z|→1-, so |Φ(0)|<1≤|Φ(z)| for all z in a neighborhood of the unit circle. So, without loss of generality, we suppose that ψ is a rotation z→eiθz and |φ(0)|<1≤|φ(z)| for all z in a neighborhood of the unit circle. Therefore, there exist a constant λ and a positive number δ<1 such that |φ(z)|<λ<1 when |z|<δ, and |φ(z)|≥1 when |z|>1-δ. Set U1={z:|z|<δ} and U2={z:|z|>1-δ}. Also, consider the sets
H1=span{Kz:z∈U1},H2=span{Kz:z∈U2},
where span {·} is the set of finite linear combinations of {·}. By using the Hahn-Banach theorem, H1 and H2 are dense subsets of H. Since ψ is a rotation, the sequence {ψnm(λ)}n is a subset of the compact set {z:|z|=λ} for each λ in 𝔻 and m=-1,1. Now by, using the Banach-Steinhaus theorem, the sequence {Kψnm(λ)}n is bounded for each λ in 𝔻 and m=-1,1. Note that, for each ℤ, |z|=|ψn(z)|. So, if z∈U1, then |φ(ψi(z))|<λ<1 and if z∈U2, then |φ(ψi-1(z))|≥1 for each positive integer i. Also, note that
Sn(Kz)=[∏i=0n-1φ(ψi(z))¯]Kψn(z)
for every positive integer n and z∈𝔻 (see [12]). Now, if z∈U1, then SnKz→0 as n→∞. Therefore the sequence {Sn} converges pointwise to zero on the dense subset H1. Define a sequence of linear maps Wn:H2→H2 by extending the definition
WnKz=[∏i=1n(φ(ψi-1(z)))-1¯]Kψn-1(z)
(z∈U2) linearly to H2. Note that, for all z∈U2, the sequence {WnKz}n is bounded and SnWnKz=Kz on H2 which implies that SnWn is identity on the dense subset H2. Hence,
‖Snf‖‖Wng‖⟶0
for every f∈H1 and every g∈H2. Now, by Theorem 2.1, the proof is complete.

Corollary 2.9.

Under the conditions of Theorem 2.8, Cφ,ψ*⨁Cφ,ψ* is supercyclic.

Proof.

It is clear since Cφ,ψ* satisfies the Supercyclicity Criterion.

Example 2.10.

Let φ(z)=(3/2)z and ψ(z)=eiθz. Then, the operator Cφ,ψ* satisfies the Supercyclicity Criterion, because 0 is an interior fixed point of ψ, and φ(0)<1≤|φ(z)| for |z|>2/3.

BayartF.MatheronE.BermúdezT.BonillaA.PerisA.On hypercyclicity and supercyclicity criteriaGethnerR. M.ShapiroJ. H.Universal vectors for operators on spaces of holomorphic functionsGodefroyG.ShapiroJ. H.Operators with dense, invariant, cyclic vector manifoldsGroße-ErdmannK.-G.Holomorphe Monster und universelle FunktionenHerreroD. A.Limits of hypercyclic and supercyclic operatorsSalasH. N.Supercyclicity and weighted shiftsYarmahmoodiS.HedayatianK.YousefiB.Supercyclicity and hypercyclicity of an isometry plus a nilpotentYousefiB.RezaeiH.Some necessary and sufficient conditions for hypercyclicity criterionYousefiB.FarrokhiniaA.On the hereditarily hypercyclic operatorsYousefiB.RezaeiH.DoroodgarJ.Supercyclicity in the operator algebra using Hilbert-Schmidt operatorsYousefiB.RezaeiH.Hypercyclic property of weighted composition operatorsYousefiB.HaghkhahS.Hypercyclicity of special operators on Hilbert function spacesYousefiB.RezaeiH.On the supercyclicity and hypercyclicity of the operator algebraYousefiB.SoltaniR.Hypercyclicity of the adjoint of weighted composition operatorsYousefiB.Hereditarily transitive tuplesto appear in Rendiconti del Circolo Matematico di Palermo10.1007/s12044-009-0048-6