# Non-Weakly Supercyclic Weighted Composition Operators

Authors: Z. Kamali; K. Hedayatian; B. Khani Robati
Journal: Abstract and Applied Analysis
 (2010)

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## 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.


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## Body

## 1. Introduction

Let𝔻 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.Forλ∈𝔻, let eλ denote the linear functional of point evaluation at λ on ℋ; that is, eλ(f)=f(λ) for every f in ℋ. Since eλ is a bounded linear functional, the Riesz representation theorem states that(1.1)eλ(f)=〈f,kλ〉
for some kλ∈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=0∞f̂(n)zn analytic on 𝔻 such that (1.2)∥f∥β2=∑n=0∞|f̂(n)|2[β(n)]2<∞.Since the function∑n=0∞(nβ(n))-1zn is in H2(β), we see that lim sup(β(n)-1/n)≤1. These spaces are Hilbert spaces with the inner product (1.3)〈f,g〉=∑n=0∞f̂(n)ĝ(n)¯[β(n)]2,
for every f and g in H2(β).Letfk(z)=zk for every nonnegative integer k. Then {fk}k is an orthogonal basis. As a particular consequence of this fact, the polynomials are dense in H2(β). It is clear that ∥fk∥=β(k). We call the set {zk/β(k),k=0,1,2,…} the standard basis for H2(β). The sequence of weights allows us to consider the generating function (1.4)k(z)=∑n=0∞znβ(n)2
which is analytic on 𝔻. Take λ∈𝔻. It is easy to see that for any function f=∑n=0∞f̂(n)zn in H2(β), (1.5)f(λ)=〈f,kλ〉,
where kλ(z)=k(λ¯z); moreover, (1.6)∥kλ∥2=∑j=0∞(|λ|2)jβ(j)2.The classical Hardy space, the Bergman space, and the Dirichlet space are weighted Hardy spaces with weights, respectively, given byβ(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.Letφ 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 ofℋ 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 Mw, defined by Mwf=wf, is bounded on ℋ. Also, note that for each λ∈𝔻, Mw*kλ=w(λ)¯kλ. 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(1.7)|w(λ)kλ(λ)|=|〈Mwkλ,kλ〉|≤∥Mw∥∥kλ∥2,
which implies that |w(λ)|≤∥Mw∥ for every λ∈𝔻, and so w∈H∞.In what follows, suppose thatw∈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 Cw,φ defined by Cw,φ(f)(z)=MwCφ(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 φn; also φ0 is the identity function. We note that (1.8)Cw,φn(f)=∏j=0n-1woφj·(foφn)
for all f∈ℋ and n≥1. Moreover, (1.9)Cw,φ*nkλ=∏j=0n-1w(φj(λ))¯kφn(λ)
for every λ∈𝔻 and n≥1. The weighted composition operators come up naturally. In 1964, Forelli [1] showed that every isometry on Hp 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 thatx∈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 is2B 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 onℋ

The following proposition limits the kinds of maps that can produce weakly supercyclic weighted composition operators onℋ.Proposition 2.1.
Suppose thatCw,φ is weakly supercyclic. Then(i)
w(λ)≠0 for all λ∈𝔻;(ii)
φ is univalent.Proof.
Suppose thatf is a weakly supercyclic vector for Cw,φ.
(i) Letλ∈𝔻, and let ɛ be a positive number. Since the set
(2.1)V={h∈ℋ:|〈h-kλ,kλ〉|<ɛ}
is a weak neighborhood of kλ, there exist n≥1 and a scalar α such that αCw,φnf∈V; that is,
(2.2)|〈αCw,φnf,kλ〉-∥kλ∥2|=|α∏j=0n-1woφj(λ)foφn(λ)-∥kλ∥2|<ɛ.
Now, if w(λ)=0, then ∥kλ∥<ɛ. Since ɛ>0 is arbitrary, we get a contradiction.
(ii) Suppose thatφ(a)=φ(b) and put
(2.3)g=(kaw(a)¯)-(kbw(b)¯).
For any ɛ>0, the set
(2.4)V={h∈ℋ:|〈h-g,g〉|<ɛ}
is a weak neighborhood of g, and so there exist n≥1 and a scalar α such that αCw,φnf∈V; that is,
(2.5)|〈αCw,φnf-g,g〉|=|〈αf,Cw,φ*ng〉-∥g∥2|=∥g∥2<ɛ.
Thus, g=0, and so 〈1,ka/w(a)¯〉=〈1,kb/w(b)¯〉; this implies that w(a)=w(b). Consequently, 〈z,ka〉=〈z,kb〉, which, in turn, implies that a=b.Hereafter, we may assume thatφ is univalent and w(z)≠0 for all z∈𝔻. Take a∈𝔻, and consider an automorphism of 𝔻 defined by ψa(z)=(a-z)/(1-a¯z),(z∈𝔻). A Hilbert space ℋ of analytic functions is called automorphism invariant if for every a∈𝔻,f∘ψa∈ℋ whenever f∈ℋ and w∘ψa∈M(ℋ) whenever w∈M(ℋ). Spaces such as the Hardy and the Bergman spaces are automorphism invariant.Theorem 2.2.
Suppose thatℋ is automorphism invariant. If Cw,φ is weakly supercyclic and λ∈𝔻 is a fixed point of φ, then (∏j=0n-1woφj(a)/(w(λ))n)n is an unbounded sequence for every a∈𝔻-{λ}.Proof.
First, suppose thatλ=0. Let f be a weakly supercyclic vector for Cw,φ. Since the constant function 1 is in ℋ, the point evaluation eλ is surjective. Therefore, the continuity of eλ implies that the set
(2.6){(αCw,φnf)(0):n≥0,α∈ℂ}={αw(0)nf(0):n≥0,α∈ℂ}
is dense in ℂ; hence, f(0) should be nonzero.
Letg(z)=z, and ɛ>0. Put
(2.7)V={h:|〈h-g,k0〉|<ɛ},W={h:|〈h-g,ka〉|<ɛ}.
Since V⋂W is a weak neighborhood of g, there exist n≥1 and a scalar α such that αCw,φnf∈V⋂W. Therefore,
(2.8)|αCw,φnf(0)-g(0)|=|αw(0)nf(0)-g(0)|<ɛ,|αCw,φnf(a)-g(a)|=|α∏j=0n-1w(φj(a))(foφn)(a)-g(a)|<ɛ.
On the other hand, since φ(0)=0, the Schwarz lemma yields that |φk(a)|<|a| for every positive integer k; so there is a constant c1 such that |f(φn(a))|≤c1 for all n. Put an=∏j=0n-1woφj(a)/(w(λ))n and assume, on the contrary, that there is a constant c2 such that |an|≤c2 for all n. Thus, (2.8) imply that
(2.9)|g(a)|≤ɛ+|αw(0)n∥an∥foφn)(a)|≤ɛ+ɛ|f(0)|c1c2
for every ɛ>0, which contradicts the fact that g(a)≠0.
Now, suppose thatλ≠0, and put
(2.10)ψλ(z)=λ-z1-λ¯z,ϕ(z)=ψλoφoψλ,W(z)=woψλ.
The operators Cw,φ and CW,ϕ are similar; indeed, Cw,φ=CψλCW,ϕCψλ-1. Thus, CW,ϕ is weakly supercyclic. If b=ψλ(a), then
(2.11)W(ϕn(b))=woψλo(ψλoφnoψλ)(b)=woφn(a)
for every positive integer n. Moreover, b≠0 and ϕ(0)=0; thus, by the previous step the sequence (W(ϕn(b))/W(0)n)n is unbounded, and so is the sequence (an)n.The following corollaries are direct consequences of Theorem2.2.Corollary 2.3.
Letℋ 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(φn(a))|≤|w(λ)| for all n≥N, then Cw,φ is not weakly supercyclic on ℋ.Example 2.4.
Suppose thatφ(z)=az, where 0<a<1 and w(z)=1-z. Then |w(φn(a))|=|1-an+1|≤|w(0)|=1 for all n≥1. Therefore, Cw,φ is not weakly supercyclic on ℋ.Corollary 2.5.
Suppose thatφ has a fixed point λ∈𝔻. If |w(λ)|≥1 and (∏j=0n-1woφj(a))n is a bounded sequence, for some a∈𝔻-{λ}; then Cw,φ is not weakly supercyclic on ℋ.Example 2.6.
Forα≠0 in 𝔻,define φ(z)=(α-z)/(1-α¯z) and w(z)=1-|α|2/(1-α¯z). Since w∈H∞ and φ(𝔻)⊆𝔻,Cw,φ is bounded on the Hardy space H2(𝔻) and the Bergman space La2(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,φj(z)=z and so w(φj(z))=w(z); moreover, for the odd values of j,φj(z)=φ(z) and w(φ(z))=1/w(z).Thus, w(φj(a))w(φj+1(a))=1 for all j≥0 and a∈𝔻. Therefore, (∏j=0n-1woφj(a))n is a bounded sequence and so Cw,φ is not weakly supercyclic.By takingw(z)≡1 in Corollary 2.3, the following result will be obtained.Corollary 2.7.
Ifℋ is automorphism invariant and φ has a fixed point in 𝔻, then Cφ is not weakly supercyclic.Proposition 2.8.
Suppose that polynomials are dense inℋ. If Cw,φ is weakly supercyclic, then w is a cyclic vector for the operator Mz.Proof.
SinceCw,φ is weakly supercyclic, its range is weakly dense. The convexity of the range of Cw,φ implies that it is norm dense, and so is the range of Mw. 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∥Mw∥. Thus,
(2.12)∥wp-g∥≤∥Mw(p-f)∥+∥wf-g∥≤∥Mw∥∥p-f∥+ϵ2<ϵ.
This shows that w is a cyclic vector for Mz.Corollary 2.9.
IfCw,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 forMz 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]).
LetK be a Hilbert space, and suppose that T is bounded operator on K. If a vector x∈K is supercyclic for T, then
(2.13)lim supn→∞|〈y,Tnx〉|∥Tnx∥∥y∥=1
for every nonzero vector y∈K.We say that a sequence(aj)j of complex numbers is not a Blaschke sequence if there exists j0 such that aj∈𝔻 for all j≥j0, and ∑j=1∞(1-|aj|) diverges.Theorem 2.11.
Suppose thatφ 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)|<|woφj(a)| for all j≥N, then Cw,φ is not supercyclic.Proof.
Ifφ is the identity map, then Cw,φ=Mw. The point spectrum of Mw* contains more than one point; in fact, Mw*kz=w(z)¯kz for all z∈𝔻. Thus, Mw is not supercyclic [24, page 13]. So, we may assume that φ is not the identity map.
Assume thatCw,φ is a supercyclic operator on ℋ and U={f∈ℋ:f(λ)≠0}. Since Uc=kereλ, the set U is open in ℋ. We know that the set of supercyclic vectors for Cw,φ is dense in ℋ [24, page 9]. So, there exists a supercyclic vector f for Cw,φ in U. If (w(λ)/w(φj(a)))j is not a Blaschke sequence, then ∏j=0∞|w(λ)/w(φj(a))|=0. On the other hand, we have
(2.14)|∏j=0n-1woφj(a)||foφn(a)|=|ea(Cw,φnf)|≤∥ea∥∥Cw,φnf∥,|(Cw,φnf)(λ)|=|∏j=0n-1woφj(λ)||f(φn(λ))|=|w(λ)|n|f(λ)|.
Therefore,
(2.15)|(Cw,φnf)(λ)|∥Cw,φnf∥≤|w(λ)|n|f(λ)|∥ea∥|∏j=0n-1woφj(a)||foφn(a)|=∏j=0n-1|w(λ)woφj(a)||f(λ)|∥ea∥|foφn(a)|.
However since φn(a)→λ and f(λ)≠0, the left side of the above inequality tends to zero, but by the angle criterion,
(2.16)lim supn→∞|(Cw,φnf)(λ)|∥Cw,φnf∥∥kλ∥=1
which is a contradiction.
Now, suppose that(w(λ)/w(φj(a)))j is a Blaschke sequence. If g(z)=z-λ, then there exists a sequence (αkCw,φnk)k such that αkCw,φnkf→g as k→∞. So
(2.17)αk∏j=0nk-1woφj(λ)foφnk(λ)=αk(w(λ))nkf(λ)→g(λ)=0,αk∏j=0nk-1woφj(a)foφnk(a)→g(a)≠0
as k→∞. Thus,
(2.18)∏j=0nk-1w(λ)woφj(a)f(λ)f(φnk(a))→0,
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]).
LetK 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
(2.19)∑n=0∞|〈y,Tnx〉|∥Tnx∥<∞.
Then, x is not a weakly supercyclic vector for T.An operatorT 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(Cw,φ)=(0); so a weighted composition operator is bounded below if and only if its range is closed.Proposition 2.13.
Suppose thatφ has a fixed point λ∈𝔻. If
(2.20)c=inf{∥Cw,φf∥:∥f∥=1}>|w(λ)|,
then Cw,φ is not weakly supercyclic.Proof.
Iff∈ℋ, then ∥Cw,φnf∥≥cn∥f∥ and
(2.21)|〈Cw,φnf,kλ〉|∥Cw,φnf∥≤|w(λ)|n|f(λ)|cn∥f∥=|w(λ)c|n|f(λ)|∥f∥.
Consequently, ∑n=1∞(|〈Cw,φnf,kλ〉|/∥Cw,φnf∥)<∞, and theorem follows by the weak angle criterion.Corollary 2.14.
Letφ 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 Cw,φ is not weakly supercyclic on the Hardy space H2(𝔻).Proof.
Sinceφ is an inner function, ∥Cφf∥≥((1-|φ(0)|)/(1+|φ(0)|))1/2∥f∥ for all f∈H2 [2, page 123]. So
(2.22)∥Cw,φf∥≥m∥foφ∥≥m1-|φ(0)|1+|φ(0)|∥f∥,
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, thenCw,φ is not weakly hypercyclic.(i)
∥Cφ∥≤1, ∥w∥≤1 and ∥k0∥≤1.(ii)
If a is the Denjoy-Wolff point ofφ, then w is continuous at a with ∥w(a)Cφ∥<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 functionφ has a fixed point in 𝔻.Proof.
Assume, on the contrary, thatCw,φ has a weakly hypercyclic vector f. If λ∈𝔻 and n≥0, then
(2.23)〈Cw,φnf,kλ〉=〈f,Cw,φ*nkλ〉=∏j=0n-1w(φj(λ))f(φn(λ)).
Suppose that (i) holds. Put λ=0 in (2.23). Since ∥kλ∥≤1, we have
(2.24)|Cw,φnf(λ)|=|〈Cw,φnf,kλ〉|=||〈Cφnf,kλ〉|∏j=0n-1〈Cφjw,kλ〉|≤∥Cφn∥∥f∥∏j=0n-1‍∥Cφ∥j∥w∥≤∥f∥.
However, the mapping eλ is weakly continuous [28, page 167], and so the sequence ((Cw,φnf)(λ))n is dense in ℂ, which is impossible.
If(ii) holds and λ∈𝔻, then by the Denjoy-Wolff theorem φn(λ)→a as n→∞. Since ∥w(a)Cφ∥<1, there is j0 such that |w(φj(λ))|∥Cφ∥<1 for all j>j0. Hence, by (2.23), we have
(2.25)|Cw,φnf(λ)|=|∏j=0n-1w(φj(λ))||〈Cφnf,kλ〉|≤∥f∥∥kλ∥|∏j=0n-1w(φj(λ))|∥Cφ∥≤∥f∥∥kλ∥|∏j=0j0w(φj(λ))|∥Cφ∥.
Thus, the sequence ((Cw,φnf)(λ))n is bounded; now, similar to part (i), we get a contradiction.
Suppose that(iii) holds. If λ∈𝔻, then φn(λ)→a and so w(φn(λ))→w(a) as n→∞. Since |w(a)|<1, ∑j=0∞(1-|w(φj(λ))|)=∞; equivalently, limn→∞∏j=0n-1w(φj(λ))=0. Now, by (2.23), the sequence ((Cw,φnf)(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 Cw,φ*kp=w(p)¯kp, the point spectrum of Cw,φ is nonempty, which is a contradiction [24, page 11].We remark that ifℋ =H2(β), then ∥k0∥=1; moreover, if lim infj→∞β(j)2/j>1, then ℋ⊆H∞. Indeed, for every f∈ℋ and λ∈𝔻, we have (2.26)|f(λ)|2=|〈f,kλ〉|2≤∥f∥2∥kλ∥2≤∥f∥2∑j-0∞1β(j)2<∞.


## 3. Weighted Composition Operators onH(𝔻)

In this section, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functionsH(𝔻). If K is a compact subset of 𝔻, for f∈H(𝔻), define PK(f)=supz∈K|f(z)|. Then {PK: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(fn)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 fn(K)∩K=∅.For example, ifφ is an automorphism, then (φn)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]).
LetX 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 Tn(U)∩V≠∅.Theorem 3.3.
If(φn)n is a run-away sequence and w is in H∞ such that w(z)≠0 for all z∈𝔻, then Cw,φ is a hypercyclic operator on H(𝔻).Proof.
LetU 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
(3.1){h∈H(𝔻):supz∈K1|h(z)-f(z)|<ϵ}⊆U,{h∈H(𝔻):supz∈K2|h(z)-g(z)|<ϵ}⊆V.
Put K=K1∪K2. Therefore,
(3.2){h∈H(𝔻):supz∈K|h(z)-f(z)|<ϵ}⊆U,{h∈H(𝔻):supz∈K|h(z)-g(z)|<ϵ}⊆V.
Since K is compact, we can choose two closed discs, B1,B2, in 𝔻 such that K⊆B1⊆B2o. Since (φn)n is a run-away sequence, there exists a positive integer n such that φn(B2)∩B2=∅. Now, consider the map R, defined by f(z), if z∈B1, and by ∏j=1n(1/woφ-j(z))goφ-n(z) if z∈φn(B1). The sets B1andφn(B1) are two disjoint closed subsets of 𝔻; moreover, since by convention, φ and so φn are injective, the complement of B1∪φn(B1) is connected. Then by Runge's theorem, there exists a polynomial p such that
(3.3)|R(z)-p(z)|<min{ϵ,ϵ∥w∥∞n}
for all z∈B1∪φn(B1). So
(3.4)|f(z)-p(z)|<ϵ∀z∈B1,
which implies that p∈U. Also,
(3.5)|∏j=1n1woφ-j(z)goφ-n(z)-p(z)|<ϵ∥w∥∞n,∀z∈φn(B1),
which implies that
(3.6)|Cw,φnp(z)-g(z)|<ϵ,∀z∈B1.
Therefore, Cw,φnp∈V. Then p∈U∩Cw,φ-n(V) which, in turn, implies that Cw,φnU⋂V is nonempty. Thus, Birkhoff's transitivity theorem implies that Cw,φ is hypercyclic.Proposition 3.4.
Suppose thatCw,φ is hypercyclic on H(𝔻) and (φn)n is not a run-away sequence. Then ranw∩∂𝔻≠∅.Proof.
Assume that for allz∈𝔻,|w(z)|<1 or for all z∈𝔻,|w(z)|>1. Let f be a hypercyclic vector for Cw,φ, and choose a subsequence (nk)k so that ∏j=0nk-1woφj·(foφnk)→1 as k→∞. Since (φn)n is not a run-away sequence there exists a compact subset K of 𝔻 such that φn(K)∩K≠∅ for every positive integer n; so one can find a sequence (ξnk)k⊆K satisfying φnk(ξnk)∈K. Since K is a compact set, there exists α∈K such that φnk(ξnk)→α (passing, if necessary, to a subsequence of (nk)), and so f(φnk(ξnk))→f(α). Now, if |w(z)|<1 for all z∈𝔻, then limk→∞(1-|w(φnk(ξnk))|)≠0 which implies that ∑j=0∞(1-|w(φj(ξnk))|)=∞ or, equivalently, ∏j=0nk-1woφj(ξnk)→0 as k→∞. In the case that |w(z)|>1 for all z∈𝔻, using a similar argument, we obtain ∏j=0nk-1((woφj)(ξnk))-1→0 as k→∞; so in both cases we have contradictions, due to the fact that ∏j=0nk-1woφj·(foφnk)→1.It is worthy of attention to note that by a similar argument as in the proof of Theorem2.2, 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 thatφ is a univalent self-map of the disc. Then the following statements are equivalent:(a)
φ has no fixed point in 𝔻.(b)
λCφ is weakly hypercyclic on H(𝔻) for every complex number λ with |λ|=1.(c)
Cφ is weakly supercyclic on H(𝔻).Proof.
If (a) holds, then by Proposition2.3 of [25], λCφ 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 thatφ is a self-map of the disc with the Denjoy-Wollf point α in 𝔻 satisfying w(α)≠0. Then Cw,φ is weakly supercyclic if and only if Cφ is weakly supercyclic on H(𝔻).Proof.
By Proposition2.6. of [25], w(α) is an eigenvalue for Cw,φ with an eigenvector g which has no zero in 𝔻; so the operator Mg is invertible. On the other hand,
(3.7)Cw,φMg=Mg(w(α)Cφ),
thus, Cw,φ is similar to w(α)Cφ. Now, the result follows from the comparison principle [24, Page 10].Corollary 3.7.
Suppose thatφ is a univalent self-map of the disc with the Denjoy-Wollf point α in 𝔻 such that |w(α)|=1. Then Cw,φ is weakly hypercyclic if and only if Cφ is weakly hypercyclic on H(𝔻).Proof.
Corollary3.5 shows that w(α)Cφ is weakly hypercyclic if and only if Cφ is weakly hypercyclic. Now, the proof is the same as the preceding proposition.



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*Source: 143808-2010-11-02.xml*