Non-Weakly Supercyclic Weighted Composition Operators

Abstract

We give sufficient conditions under which a weighted composition operator on a Hilbert space of analytic functions is not weakly supercyclic. Also, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functions on the open unit disc.

1. Introduction

Let 𝔻 denote the open unit disc in the complex plane. Let ℋ be a Hilbert space of analytic functions defined on 𝔻 such that 1,𝑧∈ℋ, and for each 𝜆 in 𝔻, the linear functional of point evaluation at 𝜆 given by 𝑓↦𝑓(𝜆) is bounded. By a Hilbert space of analytic functions ℋ we mean the one satisfying the above conditions.

For 𝜆∈𝔻, let 𝑒_𝜆 denote the linear functional of point evaluation at 𝜆 on ℋ; that is, 𝑒_𝜆(𝑓)=𝑓(𝜆) for every 𝑓 in ℋ. Since 𝑒_𝜆 is a bounded linear functional, the Riesz representation theorem states that𝑒_𝜆(𝑓)=⟨𝑓,𝑘_𝜆⟩ (1.1) for some 𝑘_𝜆∈ℋ.

A well-known example of a Hilbert space of analytic functions is the weighted Hardy space. Let (𝛽(𝑛))_𝑛 be a sequence of positive numbers with 𝛽(0)=1. The weighted Hardy space 𝐻²(𝛽) is defined as the space of functions 𝑓=∑^∞_𝑛=0𝑓(𝑛)𝑧^𝑛 analytic on 𝔻 such that ‖𝑓‖²_𝛽=^∞_𝑛=0|𝑓(𝑛)|²[𝛽(𝑛)]²<∞. (1.2)

Since the function ∑^∞_𝑛=0(𝑛𝛽(𝑛))⁻¹𝑧^𝑛 is in 𝐻²(𝛽), we see that limsup(𝛽(𝑛)^−1/𝑛)≤1. These spaces are Hilbert spaces with the inner product ⟨𝑓,𝑔⟩=^∞_𝑛=0𝑓(𝑛)𝑔(𝑛)[𝛽(𝑛)]², (1.3) for every 𝑓 and 𝑔 in 𝐻²(𝛽).

Let 𝑓_𝑘(𝑧)=𝑧^𝑘 for every nonnegative integer 𝑘. Then {𝑓_𝑘}_𝑘 is an orthogonal basis. As a particular consequence of this fact, the polynomials are dense in 𝐻²(𝛽). It is clear that ‖𝑓_𝑘‖=𝛽(𝑘). We call the set {𝑧^𝑘/𝛽(𝑘),𝑘=0,1,2,…} the standard basis for 𝐻²(𝛽). The sequence of weights allows us to consider the generating function 𝑘(𝑧)=^∞_𝑛=0𝑧^𝑛𝛽(𝑛)² (1.4) which is analytic on 𝔻. Take 𝜆∈𝔻. It is easy to see that for any function 𝑓=∑^∞_𝑛=0𝑓(𝑛)𝑧^𝑛 in 𝐻²(𝛽), 𝑓(𝜆)=⟨𝑓,𝑘_𝜆⟩, (1.5) where 𝑘_𝜆(𝑧)=𝑘(𝜆𝑧); moreover, ‖𝑘_𝜆‖²=^∞_𝑗=0(|𝜆|²)^𝑗𝛽(𝑗)². (1.6)

The classical Hardy space, the Bergman space, and the Dirichlet space are weighted Hardy spaces with weights, respectively, given by 𝛽(𝑗)=1,𝛽(𝑗)=(𝑗+1)^−1/2, and 𝛽(𝑗)=(𝑗+1)^1/2. Weighted Bergman and Dirichlet spaces are also weighted Hardy spaces.

Let 𝜑 be an automorphism of the disc. Recall that 𝜑 is elliptic if it has one fixed point in the disc and the other in the complement of the closed disc, hyperbolic if both of its fixed points are on the unit circle, and parabolic if it has one fixed point on the unit circle (of multiplicity two).

Recall that a multiplier of ℋ is an analytic function 𝑤 on 𝔻 such that 𝑤ℋ⊆ℋ. The set of all multipliers of ℋ is denoted by 𝑀(ℋ). If 𝑤 is a multiplier, then the multiplication operator 𝑀_𝑤, defined by 𝑀_𝑤𝑓=𝑤𝑓, is bounded on ℋ. Also, note that for each 𝜆∈𝔻, 𝑀^∗_𝑤𝑘_𝜆=𝑤(𝜆)𝑘_𝜆. It is known that 𝑀(ℋ)⊆𝐻^∞. In fact, since the constant functions are in ℋ, every function in 𝑀(ℋ) is analytic on 𝔻. Moreover, if 𝜆∈𝔻, then|𝑤(𝜆)𝑘_𝜆(𝜆)|=|⟨𝑀_𝑤𝑘_𝜆,𝑘_𝜆⟩|≤‖𝑀_𝑤‖‖𝑘_𝜆‖², (1.7) which implies that |𝑤(𝜆)|≤‖𝑀_𝑤‖ for every 𝜆∈𝔻, and so 𝑤∈𝐻^∞.

In what follows, suppose that 𝑤∈𝑀(ℋ) and that 𝜑 is an analytic self-map of 𝔻 such that (𝑓∘𝜑)(𝑧)=𝑓(𝜑(𝑧)) is in ℋ for every 𝑓∈ℋ. An application of the closed graph theorem shows that the weighted composition operator 𝐶_𝑤,𝜑 defined by 𝐶_𝑤,𝜑(𝑓)(𝑧)=𝑀_𝑤𝐶_𝜑(𝑓)(𝑧)=𝑤(𝑧)𝑓(𝜑(𝑧)) is bounded. The mapping 𝜑 is called the composition map and 𝑤 is called the weight. For a positive integer 𝑛, the 𝑛th iterate of 𝜑 is denoted by 𝜑_𝑛; also 𝜑₀ is the identity function. We note that 𝐶^𝑛_𝑤,𝜑(𝑓)=^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗⋅(𝑓𝑜𝜑_𝑛) (1.8) for all 𝑓∈ℋ and 𝑛≥1. Moreover, 𝐶^∗𝑛_𝑤,𝜑𝑘_𝜆=^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝜆))𝑘_{𝜑𝑛(𝜆)} (1.9) for every 𝜆∈𝔻 and 𝑛≥1. The weighted composition operators come up naturally. In 1964, Forelli [1] showed that every isometry on 𝐻^𝑝 for 1<𝑝<∞ and 𝑝≠2 is a weighted composition operator. Recently, there has been a great interest in studying composition and weighted composition operators on the unit disc, polydisc, or the unit ball; see, for example, monographs [2, 3], papers [4–18].

Suppose that 𝑥∈ℋ and 𝑇 is an operator on ℋ. The set {𝑥,𝑇𝑥,𝑇²𝑥,…}, denoted by Orb{𝑇,𝑥}, is called the orbit of 𝑥 under 𝑇. If there is a vector 𝑥∈ℋ whose orbit is (weakly) dense in ℋ, then 𝑇 is called a (weakly) hypercyclic operator and 𝑥 is called a (weakly) hypercyclic vector for 𝑇. The operator 𝑇 is called (weakly) supercyclic if the set of scalar multiples of the elements of Orb{𝑇,𝑥} is (weakly) dense and is called cyclic if the linear span of Orb{𝑇,𝑥} is dense. In each case, the vector 𝑥 is called, respectively, (weakly) supercyclic vector or cyclic vector for 𝑇.

Hypercyclicity of operators is studied a lot in literature. The classical hypercyclic operator is 2𝐵 on the space ℓ²(ℕ), where 𝐵 is the backward shift [19]. It is proved that many famous operators are hypercyclic. For instance, certain operators in the classes of composition operators [20], the adjoint of subnormal, hyponormal, and multiplication operators [21, 22] and weighted shift operators [23] are hypercyclic. As a good source on hypercyclicity, supercyclicity, weak hypercyclicity, and weak supercyclicity of operators, one can see [24].

Recently the hypercyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions is discussed in [25]. Chan and Sanders [26] showed that a weakly hypercyclic operator may fail to be norm hypercyclic. Furthermore, Sanders in [27] proved that there exists a weakly supercyclic operator that fails to be norm supercyclic. In this paper, we discuss weak supercyclicity and weak hypercyclicity of a weighted composition operator.

2. Weighted Composition Operators on ℋ

The following proposition limits the kinds of maps that can produce weakly supercyclic weighted composition operators on ℋ.

Proposition 2.1. Suppose that 𝐶_𝑤,𝜑 is weakly supercyclic. Then(i)𝑤(𝜆)≠0 for all 𝜆∈𝔻;(ii)𝜑 is univalent.

Proof. Suppose that 𝑓 is a weakly supercyclic vector for 𝐶_𝑤,𝜑.
(i) Let 𝜆∈𝔻, and let 𝜀 be a positive number. Since the set 𝑉={ℎ∈ℋ∶|⟨ℎ−𝑘_𝜆,𝑘_𝜆⟩|<𝜀} (2.1) is a weak neighborhood of 𝑘_𝜆, there exist 𝑛≥1 and a scalar 𝛼 such that 𝛼𝐶^𝑛_𝑤,𝜑𝑓∈𝑉; that is, |⟨𝛼𝐶^𝑛_𝑤,𝜑𝑓,𝑘_𝜆⟩−‖𝑘_𝜆‖²|=|𝛼^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝜆)𝑓𝑜𝜑_𝑛(𝜆)−‖𝑘_𝜆‖²|<𝜀. (2.2) Now, if 𝑤(𝜆)=0, then ‖𝑘_𝜆‖<√𝜀. Since 𝜀>0 is arbitrary, we get a contradiction.
(ii) Suppose that 𝜑(𝑎)=𝜑(𝑏) and put 𝑘𝑔=(^𝑎𝑘𝑤(𝑎))−(^𝑏𝑤(𝑏)). (2.3) For any 𝜀>0, the set 𝑉={ℎ∈ℋ∶|⟨ℎ−𝑔,𝑔⟩|<𝜀} (2.4) is a weak neighborhood of 𝑔, and so there exist 𝑛≥1 and a scalar 𝛼 such that 𝛼𝐶^𝑛_𝑤,𝜑𝑓∈𝑉; that is, |⟨𝛼𝐶^𝑛_𝑤,𝜑𝑓−𝑔,𝑔⟩|=|⟨𝛼𝑓,𝐶^∗𝑛_𝑤,𝜑𝑔⟩−‖𝑔‖²|=‖𝑔‖²<𝜀. (2.5) Thus, 𝑔=0, and so ⟨1,𝑘_𝑎/𝑤(𝑎)⟩=⟨1,𝑘_𝑏/𝑤(𝑏)⟩; this implies that 𝑤(𝑎)=𝑤(𝑏). Consequently, ⟨𝑧,𝑘_𝑎⟩=⟨𝑧,𝑘_𝑏⟩, which, in turn, implies that 𝑎=𝑏.

Hereafter, we may assume that 𝜑 is univalent and 𝑤(𝑧)≠0 for all 𝑧∈𝔻. Take 𝑎∈𝔻, and consider an automorphism of 𝔻 defined by 𝜓_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧),(𝑧∈𝔻). A Hilbert space ℋ of analytic functions is called automorphism invariant if for every 𝑎∈𝔻,𝑓∘𝜓_𝑎∈ℋ whenever 𝑓∈ℋ and 𝑤∘𝜓_𝑎∈𝑀(ℋ) whenever 𝑤∈𝑀(ℋ). Spaces such as the Hardy and the Bergman spaces are automorphism invariant.

Theorem 2.2. Suppose that ℋ is automorphism invariant. If 𝐶_𝑤,𝜑 is weakly supercyclic and 𝜆∈𝔻 is a fixed point of 𝜑, then (∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎)/(𝑤(𝜆))^𝑛)_𝑛 is an unbounded sequence for every 𝑎∈𝔻−{𝜆}.

Proof. First, suppose that 𝜆=0. Let 𝑓 be a weakly supercyclic vector for 𝐶_𝑤,𝜑. Since the constant function 1 is in ℋ, the point evaluation 𝑒_𝜆 is surjective. Therefore, the continuity of 𝑒_𝜆 implies that the set {(𝛼𝐶^𝑛_𝑤,𝜑𝑓)(0)∶𝑛≥0,𝛼∈ℂ}={𝛼𝑤(0)^𝑛𝑓(0)∶𝑛≥0,𝛼∈ℂ} (2.6) is dense in ℂ; hence, 𝑓(0) should be nonzero.
Let 𝑔(𝑧)=𝑧, and 𝜀>0. Put 𝑉={ℎ∶|⟨ℎ−𝑔,𝑘₀⟩|<𝜀},𝑊={ℎ∶|⟨ℎ−𝑔,𝑘_𝑎⟩|<𝜀}. (2.7) Since 𝑉⋂𝑊 is a weak neighborhood of 𝑔, there exist 𝑛≥1 and a scalar 𝛼 such that 𝛼𝐶^𝑛_𝑤,𝜑𝑓∈𝑉⋂𝑊. Therefore, |𝛼𝐶^𝑛_𝑤,𝜑𝑓(0)−𝑔(0)|=|𝛼𝑤(0)^𝑛𝑓(0)−𝑔(0)|<𝜀,|𝛼𝐶^𝑛_𝑤,𝜑𝑓(𝑎)−𝑔(𝑎)|=|𝛼∏^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝑎))(𝑓𝑜𝜑_𝑛)(𝑎)−𝑔(𝑎)|<𝜀. (2.8) On the other hand, since 𝜑(0)=0, the Schwarz lemma yields that |𝜑_𝑘(𝑎)|<|𝑎| for every positive integer 𝑘; so there is a constant 𝑐₁ such that |𝑓(𝜑_𝑛(𝑎))|≤𝑐₁ for all 𝑛. Put 𝑎_𝑛=∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎)/(𝑤(𝜆))^𝑛 and assume, on the contrary, that there is a constant 𝑐₂ such that |𝑎_𝑛|≤𝑐₂ for all 𝑛. Thus, (2.8) imply that |𝑔(𝑎)|≤𝜀+|𝛼𝑤(0)^𝑛‖𝑎_𝑛‖𝑓𝑜𝜑_𝑛𝜀)(𝑎)|≤𝜀+𝑐|𝑓(0)|¹𝑐₂ (2.9) for every 𝜀>0, which contradicts the fact that 𝑔(𝑎)≠0.
Now, suppose that 𝜆≠0, and put 𝜓_𝜆(𝑧)=𝜆−𝑧1−𝜆𝑧,𝜙(𝑧)=𝜓_𝜆𝑜𝜑𝑜𝜓_𝜆,𝑊(𝑧)=𝑤𝑜𝜓_𝜆. (2.10) The operators 𝐶_𝑤,𝜑 and 𝐶_𝑊,𝜙 are similar; indeed, 𝐶_𝑤,𝜑=𝐶_𝜓𝜆𝐶_𝑊,𝜙𝐶_𝜓⁻¹_𝜆. Thus, 𝐶_𝑊,𝜙 is weakly supercyclic. If 𝑏=𝜓_𝜆(𝑎), then 𝑊(𝜙_𝑛(𝑏))=𝑤𝑜𝜓_𝜆𝑜(𝜓_𝜆𝑜𝜑_𝑛𝑜𝜓_𝜆)(𝑏)=𝑤𝑜𝜑_𝑛(𝑎) (2.11) for every positive integer 𝑛. Moreover, 𝑏≠0 and 𝜙(0)=0; thus, by the previous step the sequence (𝑊(𝜙_𝑛(𝑏))/𝑊(0)^𝑛)_𝑛 is unbounded, and so is the sequence (𝑎_𝑛)_𝑛.

The following corollaries are direct consequences of Theorem 2.2.

Corollary 2.3. Let ℋ be automorphism invariant and 𝜑 have a fixed point 𝜆 in 𝔻. If there is a point 𝑎∈𝔻−{𝜆} and a positive integer 𝑁 such that |𝑤(𝜑_𝑛(𝑎))|≤|𝑤(𝜆)| for all 𝑛≥𝑁, then 𝐶_𝑤,𝜑 is not weakly supercyclic on ℋ.

Example 2.4. Suppose that 𝜑(𝑧)=𝑎𝑧, where 0<𝑎<1 and 𝑤(𝑧)=1−𝑧. Then |𝑤(𝜑_𝑛(𝑎))|=|1−𝑎^𝑛+1|≤|𝑤(0)|=1 for all 𝑛≥1. Therefore, 𝐶_𝑤,𝜑 is not weakly supercyclic on ℋ.

Corollary 2.5. Suppose that  𝜑 has a fixed point 𝜆∈𝔻. If |𝑤(𝜆)|≥1 and (∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎))_𝑛 is a bounded sequence, for some 𝑎∈𝔻−{𝜆}; then 𝐶_𝑤,𝜑 is not weakly supercyclic on ℋ.

Example 2.6. For 𝛼≠0 in 𝔻, define 𝜑(𝑧)=(𝛼−𝑧)/(1−𝛼𝑧) and 𝑤(𝑧)=1−|𝛼|²/(1−𝛼𝑧). Since 𝑤∈𝐻^∞ and 𝜑(𝔻)⊆𝔻, 𝐶_𝑤,𝜑 is bounded on the Hardy space 𝐻²(𝔻) and the Bergman space 𝐿²_𝑎(𝔻). An easy computation shows that 𝜑 is an elliptic automorphism with a fixed point 𝜆=(1−1−|𝛼|²)/𝛼 and 𝑤(𝜆)=1. Also, for the even values of j, 𝜑_𝑗(𝑧)=𝑧 and so 𝑤(𝜑_𝑗(𝑧))=𝑤(𝑧); moreover, for the odd values of j, 𝜑_𝑗(𝑧)=𝜑(𝑧) and 𝑤(𝜑(𝑧))=1/𝑤(𝑧). Thus, 𝑤(𝜑_𝑗(𝑎))𝑤(𝜑_𝑗+1(𝑎))=1 for all 𝑗≥0 and 𝑎∈𝔻. Therefore, (∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎))_𝑛 is a bounded sequence and so 𝐶_𝑤,𝜑 is not weakly supercyclic.

By taking 𝑤(𝑧)≡1 in Corollary 2.3, the following result will be obtained.

Corollary 2.7. If ℋ is automorphism invariant and 𝜑 has a fixed point in 𝔻, then 𝐶_𝜑 is not weakly supercyclic.

Proposition 2.8. Suppose that polynomials are dense in ℋ. If 𝐶_𝑤,𝜑 is weakly supercyclic, then w is a cyclic vector for the operator 𝑀_𝑧.

Proof. Since 𝐶_𝑤,𝜑 is weakly supercyclic, its range is weakly dense. The convexity of the range of 𝐶_𝑤,𝜑 implies that it is norm dense, and so is the range of 𝑀_𝑤. Suppose that 𝑔∈ℋ and 𝜖>0. Then there is a vector 𝑓 in ℋ such that ‖𝑤𝑓−𝑔‖<𝜖/2. Moreover, there is a polynomial 𝑝 such that ‖𝑝−𝑓‖≤𝜖/2‖𝑀_𝑤‖. Thus, ‖𝑤𝑝−𝑔‖≤‖𝑀_𝑤(𝑝−𝑓)‖+‖𝑤𝑓−𝑔‖≤‖𝑀_𝑤𝜖‖‖𝑝−𝑓‖+2<𝜖. (2.12) This shows that 𝑤 is a cyclic vector for 𝑀_𝑧.

Corollary 2.9. If 𝐶_𝑤,𝜑 is weakly supercyclic on the Hardy space 𝐻²(𝔻), then w is an outer function.

Proof. By Beurling's theorem, the only cyclic vectors for 𝑀_𝑧 on 𝐻²(𝔻) are the outer functions.

A necessary condition for the supercyclicity of an operator, called the angle criterion, was introduced by A. Montes-Rodríguez and H. N. Salas. They used this criterion to construct the non-supercyclic vectors for certain weighted shifts. We need the Hilbert space setting version of this criterion which is presented here, for the sake of completeness.

Theorem 2.10 (the angle criterion [24]). Let 𝐾 be a Hilbert space, and suppose that T is bounded operator on 𝐾. If a vector 𝑥∈𝐾 is supercyclic for T, then limsup_𝑛→∞|⟨𝑦,𝑇^𝑛𝑥⟩|‖𝑇^𝑛𝑥‖‖𝑦‖=1 (2.13) for every nonzero vector 𝑦∈𝐾.

We say that a sequence (𝑎_𝑗)_𝑗 of complex numbers is not a Blaschke sequence if there exists 𝑗₀ such that 𝑎_𝑗∈𝔻 for all 𝑗≥𝑗₀, and ∑^∞_𝑗=1(1−|𝑎_𝑗|) diverges.

Theorem 2.11. Suppose that 𝜑 is not an elliptic automorphism and has a fixed point 𝜆∈𝔻. If there exist 𝑎∈𝔻−{𝜆} and a positive integer 𝑁 so that |𝑤(𝜆)|<|𝑤𝑜𝜑_𝑗(𝑎)| for all 𝑗≥𝑁, then 𝐶_𝑤,𝜑 is not supercyclic.

Proof. If 𝜑 is the identity map, then 𝐶_𝑤,𝜑=𝑀_𝑤. The point spectrum of 𝑀^∗_𝑤 contains more than one point; in fact, 𝑀^∗_𝑤𝑘_𝑧=𝑤(𝑧)𝑘_𝑧 for all 𝑧∈𝔻. Thus, 𝑀_𝑤 is not supercyclic [24, page 13]. So, we may assume that 𝜑 is not the identity map.
Assume that 𝐶_𝑤,𝜑 is a supercyclic operator on ℋ and 𝑈={𝑓∈ℋ∶𝑓(𝜆)≠0}. Since 𝑈^𝑐=ker𝑒_𝜆, the set 𝑈 is open in ℋ. We know that the set of supercyclic vectors for 𝐶_𝑤,𝜑 is dense in ℋ [24, page 9]. So, there exists a supercyclic vector 𝑓 for 𝐶_𝑤,𝜑 in 𝑈. If (𝑤(𝜆)/𝑤(𝜑_𝑗(𝑎)))_𝑗 is not a Blaschke sequence, then ∏^∞_𝑗=0|𝑤(𝜆)/𝑤(𝜑_𝑗(𝑎))|=0. On the other hand, we have |^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎)||𝑓𝑜𝜑_𝑛(𝑎)|=|𝑒_𝑎(𝐶^𝑛_𝑤,𝜑𝑓)|≤‖𝑒_𝑎‖‖𝐶^𝑛_𝑤,𝜑𝑓‖,|(𝐶^𝑛_𝑤,𝜑𝑓)(𝜆)|=|∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝜆)||𝑓(𝜑_𝑛(𝜆))|=|𝑤(𝜆)|^𝑛|𝑓(𝜆)|. (2.14) Therefore, |(𝐶^𝑛_𝑤,𝜑𝑓)(𝜆)|‖𝐶^𝑛_𝑤,𝜑≤𝑓‖|𝑤(𝜆)|^𝑛|𝑓(𝜆)|‖𝑒_𝑎‖|∏^𝑛−1_𝑗=0𝑤𝑜𝜑_𝑗(𝑎)||𝑓𝑜𝜑_𝑛=(𝑎)|^𝑛−1_𝑗=0|𝑤(𝜆)𝑤𝑜𝜑_𝑗|(𝑎)|𝑓(𝜆)|‖𝑒_𝑎‖|𝑓𝑜𝜑_𝑛.(𝑎)| (2.15) However since 𝜑_𝑛(𝑎)→𝜆 and 𝑓(𝜆)≠0, the left side of the above inequality tends to zero, but by the angle criterion, limsup_𝑛→∞|(𝐶^𝑛_𝑤,𝜑𝑓)(𝜆)|‖𝐶^𝑛_𝑤,𝜑𝑓‖‖𝑘_𝜆‖=1 (2.16) which is a contradiction.
Now, suppose that (𝑤(𝜆)/𝑤(𝜑_𝑗(𝑎)))_𝑗 is a Blaschke sequence. If 𝑔(𝑧)=𝑧−𝜆, then there exists a sequence (𝛼_𝑘𝐶^𝑛𝑘_𝑤,𝜑)_𝑘 such that 𝛼_𝑘𝐶^𝑛𝑘_𝑤,𝜑𝑓→𝑔 as 𝑘→∞. So 𝛼_𝑘^𝑛𝑘−1_𝑗=0𝑤𝑜𝜑_𝑗(𝜆)𝑓𝑜𝜑_𝑛𝑘(𝜆)=𝛼_𝑘(𝑤(𝜆))^𝑛𝑘𝛼𝑓(𝜆)→𝑔(𝜆)=0,_{𝑘𝑛𝑘−1}∏_𝑗=0𝑤𝑜𝜑_𝑗(𝑎)𝑓𝑜𝜑_𝑛𝑘(𝑎)→𝑔(𝑎)≠0 (2.17) as 𝑘→∞. Thus, ^𝑛𝑘−1_𝑗=0𝑤(𝜆)𝑤𝑜𝜑_𝑗(𝑎)𝑓(𝜆)𝑓(𝜑_𝑛𝑘(𝑎))→0, (2.18) which is a contradiction.

The following theorem is a kind of the angle criterion for the weak supercyclicity which is called the weak angle criterion.

Theorem 2.12 (the weak angle criterion [24]). Let 𝐾 be a Hilbert space, and let 𝑇 be a bounded linear operator on 𝐾. Also, let 𝑥∈𝐾. Assume that one can find some nonzero 𝑦∈𝐾 such that ^∞_𝑛=0|⟨𝑦,𝑇^𝑛𝑥⟩|‖𝑇^𝑛𝑥‖<∞. (2.19) Then, x is not a weakly supercyclic vector for 𝑇.

An operator 𝑇 is bounded below if there exists 𝑐>0 such that ‖𝑇𝑓‖≥𝑐‖𝑓‖ for all 𝑓. It is well known that 𝑇 is bounded below if and only if it has a closed range and ker(𝑇)=(0). It can be easily seen that ker(𝐶_𝑤,𝜑)=(0); so a weighted composition operator is bounded below if and only if its range is closed.

Proposition 2.13. Suppose that 𝜑 has a fixed point 𝜆∈𝔻. If 𝑐=inf{‖𝐶_𝑤,𝜑𝑓‖∶‖𝑓‖=1}>|𝑤(𝜆)|, (2.20) then 𝐶_𝑤,𝜑 is not weakly supercyclic.

Proof. If 𝑓∈ℋ, then ‖𝐶^𝑛_𝑤,𝜑𝑓‖≥𝑐^𝑛‖𝑓‖ and |⟨𝐶^𝑛_𝑤,𝜑𝑓,𝑘_𝜆⟩|‖𝐶^𝑛_𝑤,𝜑≤𝑓‖|𝑤(𝜆)|^𝑛|𝑓(𝜆)|𝑐^𝑛‖𝑓‖=|𝑤(𝜆)𝑐|^𝑛|𝑓(𝜆)|.‖𝑓‖ (2.21) Consequently, ∑^∞_𝑛=1(|⟨𝐶^𝑛_𝑤,𝜑𝑓,𝑘_𝜆⟩|/‖𝐶^𝑛_𝑤,𝜑𝑓‖)<∞, and theorem follows by the weak angle criterion.

Corollary 2.14. Let 𝜑 be an inner function of the disc with a fixed point 𝜆∈𝔻. Suppose that 𝑤 is nonconstant and there exists 𝑚>0 such that |𝑤|≥𝑚 almost everywhere on 𝜕𝔻. If |𝑤(𝜆)|1+|𝜑(0)|<𝑚1−|𝜑(0)|, then 𝐶_𝑤,𝜑 is not weakly supercyclic on the Hardy space 𝐻²(𝔻).

Proof. Since 𝜑 is an inner function, ‖𝐶_𝜑𝑓‖≥((1−|𝜑(0)|)/(1+|𝜑(0)|))^1/2‖𝑓‖ for all 𝑓∈𝐻² [2, page 123]. So ‖𝐶_𝑤,𝜑𝑓‖≥𝑚‖𝑓𝑜𝜑‖≥𝑚1−|𝜑(0)|1+|𝜑(0)|‖𝑓‖, (2.22) and the result follows by applying Proposition 2.13.

In the last part of this section, we give some necessary conditions for the weak hypercyclicity of a weighted composition operator.

Proposition 2.15. If one of the following conditions holds, then 𝐶_𝑤,𝜑 is not weakly hypercyclic.(i)‖𝐶_𝜑‖≤1, ‖𝑤‖≤1 and ‖𝑘₀‖≤1.(ii)If a is the Denjoy-Wolff point of 𝜑, then 𝑤 is continuous at a with ‖𝑤(𝑎)𝐶_𝜑‖<1.(iii)ℋ⊆𝐻^∞; moreover, if 𝑎 is the Denjoy-Wolff point of 𝜑, then 𝑤 is continuous at 𝑎 and |𝑤(𝑎)|<1.(iv)The function 𝜑 has a fixed point in 𝔻.

Proof. Assume, on the contrary, that 𝐶_𝑤,𝜑 has a weakly hypercyclic vector 𝑓. If 𝜆∈𝔻 and 𝑛≥0, then ⟨𝐶^𝑛_𝑤,𝜑𝑓,𝑘_𝜆⟩=⟨𝑓,𝐶^∗𝑛_𝑤,𝜑𝑘_𝜆⟩=^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝜆))𝑓(𝜑_𝑛(𝜆)). (2.23) Suppose that (i) holds. Put 𝜆=0 in (2.23). Since ‖𝑘_𝜆‖≤1, we have |𝐶^𝑛_𝑤,𝜑𝑓(𝜆)|=|⟨𝐶^𝑛_𝑤,𝜑𝑓,𝑘_𝜆⟩|=||⟨𝐶_𝜑𝑛𝑓,𝑘_𝜆⟩|^𝑛−1_𝑗=0⟨𝐶_𝜑𝑗𝑤,𝑘_𝜆⟩|≤‖𝐶^𝑛_𝜑‖‖𝑓‖^𝑛−1_𝑗=0‖𝐶_𝜑‖^𝑗‖𝑤‖≤‖𝑓‖. (2.24) However, the mapping 𝑒_𝜆 is weakly continuous [28, page 167], and so the sequence ((𝐶^𝑛_𝑤,𝜑𝑓)(𝜆))_𝑛 is dense in ℂ, which is impossible.
If (ii) holds and 𝜆∈𝔻, then by the Denjoy-Wolff theorem 𝜑_𝑛(𝜆)→𝑎 as 𝑛→∞. Since ‖𝑤(𝑎)𝐶_𝜑‖<1, there is 𝑗₀ such that |𝑤(𝜑_𝑗(𝜆))|‖𝐶_𝜑‖<1 for all 𝑗>𝑗₀. Hence, by (2.23), we have |𝐶^𝑛_𝑤,𝜑𝑓(𝜆)|=|^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝜆))||⟨𝐶_𝜑𝑛𝑓,𝑘_𝜆⟩|≤‖𝑓‖‖𝑘_𝜆‖|^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝜆))|‖𝐶_𝜑‖≤‖𝑓‖‖𝑘_𝜆‖|^𝑗0_𝑗=0𝑤(𝜑_𝑗(𝜆))|‖𝐶_𝜑‖. (2.25) Thus, the sequence ((𝐶^𝑛_𝑤,𝜑𝑓)(𝜆))_𝑛 is bounded; now, similar to part (i), we get a contradiction.
Suppose that (iii) holds. If 𝜆∈𝔻, then 𝜑_𝑛(𝜆)→𝑎 and so 𝑤(𝜑_𝑛(𝜆))→𝑤(𝑎) as 𝑛→∞. Since |𝑤(𝑎)|<1, ∑^∞_𝑗=0(1−|𝑤(𝜑_𝑗(𝜆))|)=∞; equivalently, lim_𝑛→∞∏^𝑛−1_𝑗=0𝑤(𝜑_𝑗(𝜆))=0. Now, by (2.23), the sequence ((𝐶^𝑛_𝑤,𝜑𝑓)(𝜆))_𝑛 is bounded and similar to part (i); again we get a contradiction.
At last, suppose that (iv) holds. Let 𝑝∈𝔻 be a fixed point of 𝜑. Since 𝐶^∗_𝑤,𝜑𝑘_𝑝=𝑤(𝑝)𝑘_𝑝, the point spectrum of 𝐶_𝑤,𝜑 is nonempty, which is a contradiction [24, page 11].

We remark that if ℋ =𝐻²(𝛽), then ‖𝑘₀‖=1; moreover, if liminf_𝑗→∞𝛽(𝑗)^2/𝑗>1, then ℋ⊆𝐻^∞. Indeed, for every 𝑓∈ℋ and 𝜆∈𝔻, we have |𝑓(𝜆)|²=|⟨𝑓,𝑘_𝜆⟩|²≤‖𝑓‖²‖𝑘_𝜆‖²≤‖𝑓‖^2∞_𝑗−01𝛽(𝑗)²<∞. (2.26)

3. Weighted Composition Operators on 𝐻(𝔻)

In this section, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functions 𝐻(𝔻). If 𝐾 is a compact subset of 𝔻, for 𝑓∈𝐻(𝔻), define 𝑃_𝐾(𝑓)=sup_𝑧∈𝐾|𝑓(𝑧)|. Then {𝑃_𝐾∶𝐾⊆𝔻,𝐾iscompact} is a family of seminorms that makes 𝐻(𝔻) into a locally convex space. In fact, this topology is the topology of uniform convergence on compact subsets of the open unit disc, the usual topology for 𝐻(𝔻), which is an F-space.

Definition 3.1. A sequence of self-maps (𝑓_𝑛)_𝑛 of 𝔻 is said to be a run-away sequence if for every compact set 𝐾⊆𝔻, there exists a positive integer 𝑛 such that 𝑓_𝑛(𝐾)∩𝐾=∅.

For example, if 𝜑 is an automorphism, then (𝜑_𝑛)_𝑛 is a run-away sequence if and only if 𝜑 has no fixed point in 𝔻; see [10].

To prove the next theorem, we apply Birkhoff's transitivity theorem.

Theorem 3.2 (Birkhoff's transitivity Theorem [24]). Let 𝑋 be a separable F-space, and let 𝑇 be a bounded linear operator in 𝑋. The following statements are equivalent.(i)𝑇 is hypercyclic;(ii)𝑇 is topologically transitive; that is, for each pair of nonempty open subsets (𝑈,𝑉) of 𝑋, there exists a positive integer 𝑛 such that 𝑇^𝑛(𝑈)∩𝑉≠∅.

Theorem 3.3. If (𝜑_𝑛)_𝑛 is a run-away sequence and 𝑤 is in 𝐻^∞ such that 𝑤(𝑧)≠0 for all 𝑧∈𝔻, then 𝐶_𝑤,𝜑 is a hypercyclic operator on 𝐻(𝔻).

Proof. Let 𝑈 and 𝑉 be two nonempty open subsets of 𝐻(𝔻). There exist 𝜖>0, two compact subsets, 𝐾₁ and 𝐾₂, of 𝔻 and two functions 𝑓,𝑔∈𝐻(𝔻) such that {ℎ∈𝐻(𝔻)∶sup_𝑧∈𝐾1|ℎ(𝑧)−𝑓(𝑧)|<𝜖}⊆𝑈,{ℎ∈𝐻(𝔻)∶sup_𝑧∈𝐾2|ℎ(𝑧)−𝑔(𝑧)|<𝜖}⊆𝑉. (3.1) Put 𝐾=𝐾₁∪𝐾₂. Therefore, {ℎ∈𝐻(𝔻)∶sup_𝑧∈𝐾|ℎ(𝑧)−𝑓(𝑧)|<𝜖}⊆𝑈,{ℎ∈𝐻(𝔻)∶sup_𝑧∈𝐾|ℎ(𝑧)−𝑔(𝑧)|<𝜖}⊆𝑉. (3.2) Since 𝐾 is compact, we can choose two closed discs, 𝐵₁,𝐵₂, in 𝔻 such that 𝐾⊆𝐵₁⊆𝐵^𝑜₂. Since (𝜑_𝑛)_𝑛 is a run-away sequence, there exists a positive integer 𝑛 such that 𝜑_𝑛(𝐵₂)∩𝐵₂=∅. Now, consider the map 𝑅, defined by 𝑓(𝑧), if 𝑧∈𝐵₁, and by ∏^𝑛_𝑗=1(1/𝑤𝑜𝜑_−𝑗(𝑧))𝑔𝑜𝜑_−𝑛(𝑧) if 𝑧∈𝜑_𝑛(𝐵₁). The sets 𝐵₁𝑎𝑛𝑑𝜑_𝑛(𝐵₁) are two disjoint closed subsets of 𝔻; moreover, since by convention, 𝜑 and so 𝜑_𝑛 are injective, the complement of 𝐵₁∪𝜑_𝑛(𝐵₁) is connected. Then by Runge's theorem, there exists a polynomial 𝑝 such that 𝜖|𝑅(𝑧)−𝑝(𝑧)|<min{𝜖,‖𝑤‖^𝑛_∞} (3.3) for all 𝑧∈𝐵₁∪𝜑_𝑛(𝐵₁). So |𝑓(𝑧)−𝑝(𝑧)|<𝜖∀𝑧∈𝐵₁, (3.4) which implies that 𝑝∈𝑈. Also, |^𝑛_𝑗=11𝑤𝑜𝜑_−𝑗(𝑧)𝑔𝑜𝜑_−𝑛𝜖(𝑧)−𝑝(𝑧)|<‖𝑤‖^𝑛_∞,∀𝑧∈𝜑_𝑛(𝐵₁), (3.5) which implies that |𝐶^𝑛_𝑤,𝜑𝑝(𝑧)−𝑔(𝑧)|<𝜖,∀𝑧∈𝐵₁. (3.6) Therefore, 𝐶^𝑛_𝑤,𝜑𝑝∈𝑉. Then 𝑝∈𝑈∩𝐶^−𝑛_𝑤,𝜑(𝑉) which, in turn, implies that 𝐶^𝑛_𝑤,𝜑𝑈⋂𝑉 is nonempty. Thus, Birkhoff's transitivity theorem implies that 𝐶_𝑤,𝜑 is hypercyclic.

Proposition 3.4. Suppose that 𝐶_𝑤,𝜑 is hypercyclic on 𝐻(𝔻) and (𝜑_𝑛)_𝑛 is not a run-away sequence. Then 𝑟𝑎𝑛𝑤∩𝜕𝔻≠∅.

Proof. Assume that for all 𝑧∈𝔻,|𝑤(𝑧)|<1 or for all 𝑧∈𝔻,|𝑤(𝑧)|>1. Let 𝑓 be a hypercyclic vector for 𝐶_𝑤,𝜑, and choose a subsequence (𝑛_𝑘)_𝑘 so that ∏^𝑛𝑘−1_𝑗=0𝑤𝑜𝜑_𝑗⋅(𝑓𝑜𝜑_𝑛𝑘)→1 as 𝑘→∞. Since (𝜑_𝑛)_𝑛 is not a run-away sequence there exists a compact subset 𝐾 of 𝔻 such that 𝜑_𝑛(𝐾)∩𝐾≠∅ for every positive integer 𝑛; so one can find a sequence (𝜉_𝑛𝑘)_𝑘⊆𝐾 satisfying 𝜑_𝑛𝑘(𝜉_𝑛𝑘)∈𝐾. Since 𝐾 is a compact set, there exists 𝛼∈𝐾 such that 𝜑_𝑛𝑘(𝜉_𝑛𝑘)→𝛼 (passing, if necessary, to a subsequence of (𝑛_𝑘)), and so 𝑓(𝜑_𝑛𝑘(𝜉_𝑛𝑘))→𝑓(𝛼). Now, if |𝑤(𝑧)|<1 for all 𝑧∈𝔻, then lim_𝑘→∞(1−|𝑤(𝜑_𝑛𝑘(𝜉_𝑛𝑘))|)≠0 which implies that ∑^∞_𝑗=0(1−|𝑤(𝜑_𝑗(𝜉_𝑛𝑘))|)=∞ or, equivalently, ∏^𝑛𝑘−1_𝑗=0𝑤𝑜𝜑_𝑗(𝜉_𝑛𝑘)→0 as 𝑘→∞. In the case that |𝑤(𝑧)|>1 for all 𝑧∈𝔻, using a similar argument, we obtain ∏^𝑛𝑘−1_𝑗=0((𝑤𝑜𝜑_𝑗)(𝜉_𝑛𝑘))⁻¹→0 as 𝑘→∞; so in both cases we have contradictions, due to the fact that ∏^𝑛𝑘−1_𝑗=0𝑤𝑜𝜑_𝑗⋅(𝑓𝑜𝜑_𝑛𝑘)→1.

It is worthy of attention to note that by a similar argument as in the proof of Theorem 2.2, one can show that it can also be applied on the space 𝐻(𝔻) with a further assumption that 𝑤(𝜆)≠0. Since the space 𝐻(𝔻) is automorphism invariant, we have the following result.

Corollary 3.5. Suppose that 𝜑 is a univalent self-map of the disc. Then the following statements are equivalent:(a)𝜑 has no fixed point in 𝔻.(b)𝜆𝐶_𝜑 is weakly hypercyclic on 𝐻(𝔻) for every complex number 𝜆 with |𝜆|=1.(c)𝐶_𝜑 is weakly supercyclic on 𝐻(𝔻).

Proof. If (a) holds, then by Proposition 2.3 of [25], 𝜆𝐶_𝜑 is hypercyclic on 𝐻(𝔻) for every complex number 𝜆 with |𝜆|=1. Clearly (b) implies (c) and by the preceding discussion (c) implies (a).

Proposition 3.6. Suppose that 𝜑 is a self-map of the disc with the Denjoy-Wollf point 𝛼 in 𝔻 satisfying 𝑤(𝛼)≠0. Then 𝐶_𝑤,𝜑 is weakly supercyclic if and only if 𝐶_𝜑 is weakly supercyclic on 𝐻(𝔻).

Proof. By Proposition 2.6. of [25], 𝑤(𝛼) is an eigenvalue for 𝐶_𝑤,𝜑 with an eigenvector 𝑔 which has no zero in 𝔻; so the operator 𝑀_𝑔 is invertible. On the other hand, 𝐶_𝑤,𝜑𝑀_𝑔=𝑀_𝑔(𝑤(𝛼)𝐶_𝜑), (3.7) thus, 𝐶_𝑤,𝜑 is similar to 𝑤(𝛼)𝐶_𝜑. Now, the result follows from the comparison principle [24, Page 10].

Corollary 3.7. Suppose that 𝜑 is a univalent self-map of the disc with the Denjoy-Wollf point 𝛼 in 𝔻 such that |𝑤(𝛼)|=1. Then 𝐶_𝑤,𝜑 is weakly hypercyclic if and only if 𝐶_𝜑 is weakly hypercyclic on 𝐻(𝔻).

Proof. Corollary 3.5 shows that 𝑤(𝛼)𝐶_𝜑 is weakly hypercyclic if and only if 𝐶_𝜑 is weakly hypercyclic. Now, the proof is the same as the preceding proposition.

Acknowledgment

This research was in part supported by a grant from Shiraz University Research Council.