Weighted Composition Operators and 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 {n_k} of positive integers, and also there exist mappings $$ such that

• (1)

  $$  for every z ∈ Z,

• (2)

  $$ for every y ∈ Y and every z ∈ Z.

Then, T is supercyclic.

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:

• (1)

  T has a dense range,

• (2)

  T is supercyclic,

• (3)

  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 = ℂ  orb(T, h₀) and Y₀ = the generalized kernel of T. Since T is supercyclic, there exist sequences $$, $$ and $$ such that f_j → 0 and $$. Define $$ by

Then, clearly, $$ pointwise on X₀ and 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 C_ψ is bounded on H. We define the iterates ψ_n = ψ∘ψ∘⋯∘ψ (n times). By $$ or ψ_−n we mean the nth iterate of ψ⁻¹, hence $$ for m = −1,1.

Definition 2.5. We say that $$ is a B-sequence for ψ if ψ(z_k) = z_k−1 for all k ≥ 1.

Corollary 2.6. Suppose that $$ is a B -sequence for ψ and has limit point in 𝔻. If φ(z₀) = 0, then $$ satisfies the Supercyclicity Criterion.

Proof. Put A = C_φ,ψ. Since φ(z₀) = 0, we get $$ for all i = 0, …, n − 1. Hence A^* has dense generalized kernel. Now let $$ for all n, thus φ(z_n) · f∘ψ(z_n) = 0 for all n. This implies that f is the zero constant function, because φ is nonconstant and $$ has limit point in 𝔻  . Thus, A^* has dense range and, by Theorem 2.3, the proof is complete.

Example 2.7. Let $$, φ(z) = z − (1/2), and define $$ for all n ≥ 0. Now by Corollary 2.6, the operator $$ 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 $$ satisfies the Supercyclicity Criterion.

Proof. Put Ψ = α_p∘ψ∘α_p and Φ = φ∘α_p where

Since Ψ is an automorphism with Ψ(0) = 0, thus Ψ is a rotation z → e^iθz for some θ ∈ [0,2π] and every z ∈ U. Set $$ and $$. Then, clearly $$, thus T is similar to S which implies that S satisfies the Supercyclicity Criterion if and only if T satisfies the Supercyclicity Criterion. Since |α_p(z) | → 1⁻ when |z | → 1⁻, so |Φ(0) | < 1≤|Φ(z)| for all z in a neighborhood of the unit circle. So, without loss of generality, we suppose that ψ is a rotation z → e^iθz and |φ(0) | < 1≤|φ(z)| for all z in a neighborhood of the unit circle. Therefore, there exist a constant λ and a positive number δ < 1 such that |φ(z) | < λ < 1 when |z | < δ, and |φ(z) | ≥ 1 when |z | > 1 − δ. Set U₁ = {z:|z | < δ} and U₂ = {z:|z | > 1 − δ}. Also, consider the sets where span {·} is the set of finite linear combinations of {·}. By using the Hahn-Banach theorem, H₁ and H₂ are dense subsets of H. Since ψ is a rotation, the sequence $$ 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 $$ is bounded for each λ in 𝔻 and m = −1,1. Note that, for each ℤ, |z | = |ψ_n(z)|. So, if z ∈ U₁, then |φ(ψ_i(z)) | < λ < 1 and if z ∈ U₂, then $$ for each positive integer i. Also, note that for every positive integer n and z ∈ 𝔻 (see [12]). Now, if z ∈ U₁, then SⁿK_z → 0 as n → ∞. Therefore the sequence {Sⁿ} converges pointwise to zero on the dense subset H₁. Define a sequence of linear maps W_n : H₂ → H₂ by extending the definition (z ∈ U₂) linearly to H₂. Note that, for all z ∈ U₂, the sequence $$ is bounded and SⁿW_nK_z = K_z on H₂ which implies that SⁿW_n is identity on the dense subset H₂. Hence, 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, $$ is supercyclic.

Proof. It is clear since $$ satisfies the Supercyclicity Criterion.

Example 2.10. Let φ(z) = (3/2)z and ψ(z) = e^iθz. Then, the operator $$ satisfies the Supercyclicity Criterion, because 0 is an interior fixed point of ψ, and φ(0) < 1≤|φ(z)| for |z | > 2/3.