Compactness and Hypercyclicity of Composition. Operators between Weighted Hardy Spaces

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Compactness and Hypercyclicity of Composition

Operators between Weighted Hardy Spaces

Bahmann Yousefi

Department of Mathematics, College of Sciences, Shiraz University, 71454, Iran Shiraz Payame-Noor University, Bolvar Iman Jonubi 71345-1774, Shiraz, Iran

byouseﬁ@shirazu.ac.ir Maryam Ahmadian

Department of Mathematics, Payame-Noor University Shiraz, Bolvar Iman Jonubi 71345-1774, Shiraz, Iran

[email protected] ”Dedicated to Mola Ali”

Abstract

In this paper we investigate the necessary and suﬃcient conditions on a symbol ϕ for the boundedness, compactness and hypercyclicity of the induced composition operator C_ϕ acting between the weighted Hardy spaces Hp(β₁) and Hq(β₂), for 1< q ≤p <∞. Also, under the condition of compactness for C_ϕ we investigate the ﬁxed points ofC_ϕ.

Mathematics Subject Classification: 47B37, 47A16, 47B33

Keywords: Banach spaces of formal power series associated with a se-quenceβ, composition operator, compact operator, hypercyclic operator, Denjoy-Wolﬀ point

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1 Introduction

Let {β(n)}_n be a sequence of positive numbers withβ(0) = 1 and let 1< p <

∞. Let f ={fˆ(n)}∞_n₌₀ be such that

fp =fp_Hp₍_β₎ = +∞ n=0 |_fˆ₍_n₎_|p_β₍_n₎p _<_∞_. The notation f(z) = +∞ n=0 ˆ

f(n)zn shall be used weather or not the series con-verges for any value of z. The space of such formal power series is called the weighted Hardy space, which is denoted by Hp(β). In the casep= 2, the clas-sical Hardy space, Bergman space and the Dirichlet space are weighted Hardy spaces with β(n) = 1, β(n) = (n+ 1)−12 andβ(n) = (n+ 1)12, respectively. The space H2(β) becomes a Hilbert space with inner product

f, g= +∞ n=0 a_nb_nβ(n)2 where f(z) = +∞ n=0 a_nzn and g(z) = +∞ n=0

b_nzn are the elements of H2(β) ([1, 5]). Generally, the spaces Hp(β) are reﬂexive Banach spaces with the norm

._β and the dual of Hp(β) is Hq(βpq_{) where} 1

p +

1

q = 1 and β

p

q ₌ {_β₍_n₎pq}_n ([6]). Let ˆf_k(n) = δ_k(n). So f_k(z) = zk and then {f_k} is a basis such that

f_k=β(k). If lim β(n+ 1)

β(n) = 1 or lim infβ(n) 1

n = 1, then Hp(β) consists of functions analytic on the open unit disk U.

Remember that a complex number λ is said to be a bounded point evalu-ation on Hp(β) if the functional of point evaluation at λ, e_λ, is bounded.

We denote the set of multipliers{ϕ∈ Hp(β) :ϕHp(β)⊆Hp(β)}byH_∞p (β) and the operator of multiplication by ϕ onHp(β) byM_ϕ with ϕ_∞=M_ϕ. It is convenient and helpful to introduce the notation < f, g >to stand for

g(f) where f ∈Hp(β) and g ∈Hp(β)∗. Note that

< f, g >= ∞ n=0 ˆ f(n) ˆg(n)β(n)p.

Let ϕ be an analytic self map of U. A composition operator C_ϕ maps an analytic functionf ∈Hp(β) into (C_ϕf)(z) = f(ϕ(z)). The functionϕis called

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the composition map. Some sources on formal power series and composition operators include [1–15].

LetX be a complex Banach space andB(X) be the set of bounded linear operators from X into itself. If T ∈B(X), then the orbit of a vector x∈X is the setOrb(T, x) ={Tnx:n ∈N∪{0}}. The operator T is called hypercyclic if Orb(T, x) is dense in X for some x∈X.

2 Main Results

This work investigates the boundedness and compactness of the composition operator acting between the weighted Hardy spaces Hp(β₁) andHq(β₂). Also, we study the hypercyclicity of the composition operator C_ϕ acting between weighted Hardy spaces in the unit disc. For similar discussion on the Hardy space H2 see [4].

Theorem 1.

Suppose that ϕ is an analytic self-map of the open unit disc and the composition operator C_ϕ : Hp(β₁) → Hq(β₂) is deﬁned where 1 < q ≤ p < ∞ and β₁(n) ≤ β₂(n) for all n. Also, let 1_p + _p1 = 1 and

∞

n=0

1

β1(n)p < ∞. If ||f ◦ϕ||Hq(β2) ≤ c||f ◦ϕ||U for all f in Hp(β1), then the operator C_ϕ is bounded.

Proof.

By the property ∞ n=0

1

β1(n)p < ∞, the spaces Hp(β1) and Hq(β2) consists of functions analytic in the open unit disc U that are continuous on the unit circle. Let f ∈Hp(β₁), then C_ϕf =f◦ϕ ∈Hq(β₂) and so

||f◦ϕ||_Hq₍_β2₎ ≤c||f◦ϕ||_U ≤c||f||_U, since ϕ(U)⊆U. On the other hand, note that if

f(z) = ∞ n=0 ˆ f(n)zn ∈Hp(β₁),

then by the H¨◦lder inequality we have

|f(z)| = | ∞ n=0 ˆ f(n)zn|=| ∞ n=0 ( ˆf(n)β₁(n)) z n β₁(n)|

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≤ ( ∞ n=0 |_fˆ₍_n₎_|p_β 1(n)p)1/p.( ∞ n=0 |z|np β₁(n)p) 1/p ≤ ||f||_Hp₍_β1₎( ∞ n=0 1 β₁(n)p) 1/p

for all z inU. Hence for all f in Hp(β₁),||f||_U ≤α||f||_Hp₍_β1₎ where

α= ( ∞ n=0 1 β₁(n)p) 1/p . Now we get ||C_ϕf||=||f ◦ϕ||_Hq₍_β2₎ ≤αc||f||_Hp₍_β1₎ which implies that C_ϕ :Hp(β₁)→ Hq(β₂) is bounded. 2

Theorem 2.

Let ϕ satisﬁes the conditions of Theorem 1 and suppose that for all disc-automorphism ψ, the composition operator C_ψ : Hp(β₁) →

Hp(β₁) is deﬁned and ||f ◦ ψ||_Hp₍_β1₎ ≤ d||f ◦ ψ||_U for all f in Hp(β₁). If

C_ϕ : Hp(β₁) → Hq(β₂) is compact, then ϕ has exactly one ﬁxed point in U

and ||ϕ||_U is strictly smaller than 1.

Proof.

By Theorem 1, C_ϕ is bounded. Since ∞ n=0

1

β1(n)p <∞, each point

of ¯U is a bounded point evaluation for both spacesHp(β₁) andHq(β₂). LetC_ϕ

be compact fromHp(β₁) into Hq(β₂) and let λbe the Denjoy-Wolﬀ point ofϕ

in ¯U. Hence ϕ₍_n₎(z)→λ for allz in ¯U (here ϕ₍_n₎ is the nth iterate ofϕ). Now ifwis another ﬁxed point ofϕin ¯U, then we have 0 = lim

n (ϕ(n)(w)−λ) =w−λ and so w=λ. This implies that λ is unique.

Now if |λ| = 1, let ψ be a disc-automorphism such that ψ(ϕ(0)) = 0 and

ψ(λ) =λ. Since||f◦ψ||_Hp₍_β1₎ ≤d||f◦ψ||_U for all f in Hp(β₁),by the proof of Theorem 1, we can see that the operator C_ψ : Hp(β₁) → Hp(β₁) is bounded. Put h = ψ ◦ϕ, then h has two ﬁxed points 0 and λ in ¯U. But C_h = C_ϕC_ψ

and C_ϕ is compact, hence C_h : Hp(β₁) → Hq(β₂) is also compact that is a contradiction since in this casehmust have exactly one ﬁxed point in ¯U. Thus

λ ∈U.

Suppose that ||ϕ||_U = 1. Then there exists z₀ ∈ ∂U such that |ϕ(z₀)| =

||ϕ||_U = 1 since C_ϕf₁ =ϕ∈Hq(β₂) and Hq(β₂)⊂H(U)∩C( ¯U). Letz₀ =eiθ1

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ψ(z₀) = z₀ ∈ ∂U and C_ψ = C_ϕC_h is compact that is a contradiction. Hence

||φ||_U <1 and so the proof is complete. 2

Proposition 3.

Letβ_i(n+ 1)/β_i(n)→1 as n → ∞ for i= 1,2. Also, let β₁(n)≤β₂(n) for all n and 1< q≤p <∞. If ϕ is an analytic self-map of

U that ﬁxes a point of U and C_ϕ : Hp(β₁) → Hq(β₂) is bounded, then C_ϕ is not hypercyclic.

Proof.

First note that sinceβ_i(n+ 1)/β_i(n)→1, Hp(β₁) and Hq(β₂) are Banach spaces of functions analytic inU and each point ofU is a bounded point evaluation for both spacesHp(β₁) andHq(β₂). The functional of evaluation at

ω (w ∈ U) on both spaces Hp(β₁) and Hq(β₂) is denoted respectively by e(_wp)

and e(_wq). Now suppose α ∈U is a ﬁxed point for ϕ. Then < f, e(_αp) >=f(α) for all f in Hp(β₁). Fix f ∈ Hp(β₁), to be regarded as a hypercyclic vector candidate.

Ifg belongs to the closure ofOrb(C_ϕ, f), then for some subsequence n_k →

+∞ we have C_ϕ(_nk₎f →g. Thus we have

g(α) =< g, e(_αq)> = lim k < Cϕ(nk)f, e (q) α >= lim_k < f, Cϕ∗(_nk)e (q) α > = lim k < f, e (p) ϕ(_nk)(α)>= lim_k f(ϕ(nk)(α)) = f(α). This implies that no orbit can be dense inHq(β₂) and soC_ϕ :Hp(β₁)→Hq(β₂) is not hypercyclic. This completes the proof. 2

Theorem 4. Suppose that β_i(n+ 1)/β_i(n) → 1 as n → ∞ for i = 1,2. Let β₁(n)≤β₂(n) for alln, and 1< q ≤p <∞. Also, let ∞_n₌₀ 1

β1(n)p = +∞ where 1_p+_p1 = 1. Ifϕ is a conformal automorphism of the unit discU with no ﬁxed point inU andC_ϕ :Hp(β₁)→Hq(β₂) is bounded, thenC_ϕ is hypercyclic.

Proof.

Let α ∈ ∂U be the unique fixed point of ϕ that comes in the Denjoy-Wolff Theorem ([3]) and denote the other fixed point by β. Then this too must lie on the unit circle ∂U. Let Y_α denotes the set of polynomials that vanish at α. Since ϕ₍_n₎ → α uniformly on compact subsets of U, we get

C_ϕ(_n₎p = p◦ϕ₍_n₎ → 0 for all p in Y_α. Thus C_ϕ(_n₎ → 0 on Y_α which is dense in Hp(β₁). Put S =C_ϕ−1 and letY_β be the set of all continuous functions on the closed unit disc that are analytic in U and vanish at β. Then Y_β is dense in Hp(β₁) and S maps the dense set Y_β into itself and Sn → 0 on Y_β. The

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hypothesis of the Hypercyclicity Criterion ([14]) are therefore satisﬁed and so

C_ϕ :Hp(β₁)→Hq(β₂) is hypercyclic. 2

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