# On the Norm of Certain Weighted Composition Operators on the Hardy Space

Authors: M. Haji Shaabani; B. Khani Robati
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We obtain a representation for the norm of certain compact weighted
composition operatorCψ,φ on the Hardy space H2, whenever φ(z)=az+b and ψ(z)=az−b. We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on φ and ψ. Moreover, we characterize the norm and essential norm of such operators in a special case.


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## Body

## 1. Introduction

LetD denote the open unit disk in the complex plane. The Hardy space H2 is the space of analytic functions on D whose Taylor coefficients, in the expansion about the origin, are square summable. Also we recall that H∞ is the space of all bounded analytic function defined on D. For α∈D, the reproducing kernel at α for H2 is defined by Kα(z)=1/(1-α̅z). An easy computation shows that 〈f,Kα〉=f(α) whenever f∈H2. For any analytic self-map φ of D, the composition operator Cφ on H2 is defined by the rule Cφ(f)=f∘φ. Every composition operator is bounded, with(1.1)11-|φ(0)|2≤∥Cφ:H2→H2∥≤1+|φ(0)|1-|φ(0)|
(see [1]). We see from expression (1.1) that ∥Cφ∥=1 whenever φ(0)=0. There are  few other cases for which the exact value of the norm has been known for many years. For example, the norm of Cφ was obtained by Nordgren in [2], whenever φ is an inner function. In [3] this norm was determined, when φ(z)=az+b, with |a|+|b|≤1, and if 0<s<1 and 0≤r≤1 the norm was found in [4] for φ(z)=((r+s)z+(1-s))/(r(1-s)z+(1+rs)).In 2003, Hammond [5] obtained exact values for the norms of composition operators Cφ for certain linear fractional maps φ. In [6], Bourdon et al. determined the norm of Cφ for a large class of linear-fractional maps, including those of the form φ(z)=b/(d-z), where 0<b<d-1. The connection between the norm of certain composition operators Cφ with linear-fractional symbol acting on the Hardy space and the roots of associated hypergeometric functions was first made by Basor and Retsek [7]. It was later refined by Hammond [8]. In [9] Effinger-Dean et al. computed the norms of composition operators with rational symbols that satisfy certain properties. Their work is based on the initial work of Hammond [5]. Some other recent results regarding the calculation of the operator norm of some composition operators on the other spaces can be found in [10–14].Ifψ is a bounded analytic function on D and φ is an analytic map from D into itself, the weighted composition operator Cψ,φ is defined by Cψ,φ(f)(z)=ψ(z)f(φ(z)). The map φ is called the composition map and ψ is called the weight. If ψ is a bounded analytic function on D, then the operator can be rewritten as Cψ,φ=MψCφ, where Mψ is a multiplication operator and Cφ is a composition operator. Recall that if φ is an analytic self-map of D, then the composition operator Cφ on H2 is bounded, hence in this case Cψ,φ is bounded, but in general every weighted composition operator Cψ,φ on H2 is not bounded. If Cψ,φ is bounded, then Cψ,φ(1)=ψ belongs to H2. These operators come up naturally. In 1964, Forelli [15] 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 weighted composition operators in the unit disk, polydisk, or the unit ball; see [12, 16–27], and the references therein. In this paper we investigate the norm of certain bounded weighted composition operators Cψ,φ on H2.


## 2. Norm Calculation

In this section we obtain a representation for the norm of a class of compact weighted composition operatorsCψ,φ on the Hardy space H2, whenever φ(z)=az+b, ψ(z)=az-b, |b|2≥1/2, and 2|a|2+|b|2≤2/3. Also we give the norm and essential norm inequality for a class of noncompact weighted composition operators  Cψ,φ on H2 when φ(z)=azn+b, for some n∈ℕ, |a|+|b|=1, and ψ is a bounded analytic map on D such that the radial limit of |ψ| at one of the nth roots of b|a|/a|b| is the supremum of |ψ| on D. Also, when n=1 we obtain the norm and essential norm of such operators.The following lemma was inspired by a similar result for unweighted composition operators [28, Theorem  1.4]. See [29] for a similar proof.Lemma 2.1.
LetKw be the reproducing kernel at  w. Then
(2.1)Cψ,φ*Kw=ψ(w)¯Kφ(w).
In the next proposition we generalize the lower bound in (1.1).Proposition 2.2.
Letφ  be a nonconstant analytic self-map of  D, and let  ψ be a nonzero analytic map on  D. If n is the smallest nonnegative integer such that  ψ(n)(0)≠0, then
(2.2)∥Cψ,φ∥≥|ψ(n)(0)n!|11-|φ(0)|2.Proof.
We note that iff is in H2, then for every n∈ℕ∪{0} we have |f(n)(0)/n!|≤∥f∥2. Hence we have
(2.3)∥Cψ,φ∥≥∥Cψ,φKφ(0)∥∥Kφ(0)∥=∥ψ.(Kφ(0)∘φ)∥∥Kφ(0)∥≥|(ψ(n)(0)/n!)(Kφ(0)∘φ)(0)|∥Kφ(0)∥=|ψ(n)(0)n!|11-|φ(0)|2.LetT be a bounded operator on a Hilbert space H. We recall that ∥T∥e, the essential norm of  T, is the norm of its equivalence class in the Calkin algebra. Since the spectral radius of the operator T*T equals ∥T*T∥=∥T∥2, we study the spectrum of T*T when trying to determine ∥T∥. We say that the operator T is norm-attaining if there is a nonzero h∈H such that ∥T(h)∥=∥T∥∥h∥. We know that ∥T(h)∥=∥T∥∥h∥ if and only if T*T(h)=∥T∥2h. Moreover, if ∥T∥e<∥T∥, then the operator  T is norm-attaining and so the quantity ∥T∥2 equals the largest eigenvalue of T*T; see [5] for more details. If φ(z)=az+b, ψ(z)=az-b, |b|2≥1/2, and 2|a|2+|b|2≤2/3, then the operator Cψ,φ is compact (see the proof of Proposition 2.5). Hence 0=∥Cψ,φ∥e<∥Cψ,φ∥ and so Cψ,φ is norm-attaining.Now our goal is to find a functional equation that relates an eigenvalue ofCψ,φ*Cψ,φ to the values of its eigenfunctions at particular points in the disk. In what follows we use the techniques used in [5, 6, 30] and present some results that help us to obtain the norm of Cψ,φ.Letφ be an analytic self-map of D and let  ψ be a bounded analytic map on  D. Then
(2.4)(Cψ,φ)*=(MψCφ)*=Cφ*Mψ*=Cφ*Tψ*.But ifφ(z)=az+b such that |a|+|b|≤1, then by [3] or [28]
(2.5)(Cψ,φ)*=TgCσTh*Tψ*=TgCσ(Tψh)*,whereh(z)=1, g(z)=1/-b¯z+1,  and  σ(z)=a¯z/-b¯z+1.From now on, unless otherwise stated, we assume thatψ(z)=cz+d,φ(z)=az+b, and |a|+|b|≤1. Since Tz* is the backward shift on  H2, we see that
(2.6)Cψ,φ*Cψ,φf(z)=TgCσTψ*TψCφf(z)=TgCσTcz+d*(ψ·f(φ(z)))=TgCσ(c¯(ψ·f(φ(z))-ψ·f(φ(0))z))+TgCσ(d¯ψ(z)·f(φ(z)))=Tg(c¯(ψ(σ(z))·f(φ(σ(z)))-ψ(0)·f(φ(0))σ(z)))+d¯g(z)ψ(σ(z))·f(φ(σ(z)))=g(z)(c¯(ψ(σ(z))·f(φ(σ(z)))-ψ(0)·f(φ(0))σ(z)))+d¯g(z)ψ(σ(z))·f(φ(σ(z)))=γ(z)f(τ(z))+χ(z)f(φ(0))
for all z in D not equal to 0, where
(2.7)γ(z)=(c¯(1-b¯z)+d¯a¯z)(d(1-b¯z)+a¯cz)(a¯z)(1-b¯z)2,τ(z)=(|a|2-|b|2)z+b-b¯z+1,χ(z)=-c¯da¯z.In particular, ifg is an eigenfunction for Cψ,φ*Cψ,φ corresponding to an eigenvalue λ, then
(2.8)λg(z)=γ(z)g(τ(z))+χ(z)g(φ(0)).
Formula (2.8) is essentially identical to [5, Formula (3.3)]. Using (2.8) we can find a set of conditions under which we determine ∥Cψ,φ*Cψ,φ∥. In the trivial case a=0 we have ∥Cψ,φ∥=∥ψ∥2(1/1-|b|2). Also if d=0, then ∥Cψ,φ∥=|c|∥Cφ∥ and if c=0, then ∥Cψ,φ∥=|d|∥Cφ∥. Therefore we assume that a,b,c,d are nonzero.Throughout this paper, we writeτ[j] to denote the jth iterate of τ, that is, τ[0] is the identity map on D and τ[j+1]=τ∘τ[j].By a similar argument as in the proof of [5, Proposition   5.1], we have the following lemma.Lemma 2.3.
Letg be an eigenfunction for Cψ,φ*Cψ,φ corresponding to an eigenvalue λ, z∈D and for each nonnegative integer j, τ[j](z)≠0. Then one has
(2.9)λj+1g(z)=g(τ[j+1](z))∏k=0j[γ(τ[k](z))]+∑k=0j[g(φ(0))χ(τ[k](z))∏m=0k-1[γ(τ[m](z))]]λj-k,
where one takes ∏m=0-1(·)=1.Lemma 2.4.
For eachn∈ℕ, τ[n](0)=αnb, where {αn} is strictly increasing sequence such that αn≥1 for each n∈ℕ. Also αn+1=1+αn|a|2/(1-αn|b|2).Proof.
(By induction) Sinceτ(0)=b and τ[2](0)=(1+|a|2/(1-|b|2))b,  the claim holds for  n=1. Assume the claim holds for n-1. We will prove it for n. We have
(2.10)τ[n](0)=τ(τ[n-1](0))=τ(αn-1b)=(1+αn-1|a|21-αn-1|b|2)b.
Now if we set  αn=1+(αn-1|a|2)/(1-αn-1|b|2), then τ[n](0)=αnb. But by hypothesis αn-1<αn, so
(2.11)1+αn-1|a|21-αn-1|b|2<1+αn|a|21-αn|b|2,
which implies that αn<αn+1 also τ[n+1](0)=τ(αnb)=(1+αn|a|2/(1-αn|b|2))b. Hence the proof is complete.Proposition 2.5.
Leta=c, b=-d and let λ=∥Cψ,φ∥2. If |b|2≥1/2, and 2|a|2+|b|2≤2/3, then for each z∈D with the property that τ[j](z)≠0 for every nonnegative integer j, one has
(2.12)g(z)=∑k=0∞[g(φ(0))χ(τ[k](z))∏m=0k-1[γ(τ[m](z))]]1λk+1.Proof.
Since2|a|2+|b|2≤2/3, it is easy to see that |a|+|b|=1 if and only if |a|=1/3 and |b|=2/3. By assumption  |b|2≥1/2, so |a|+|b|<1. Therefore Cφ is compact and, since Cψ,φ=MψCφ, the operator Cψ,φ is compact. Now according to the paragraph after Proposition 2.2, there is function g in H2 such that Cψ,φ*Cψ,φg=λg. Let z∈D and for each integer j≥0, τ[j](z)≠0. By Lemma 2.3, we have
(2.13)λj+1g(z)=g(τ[j+1](z))∏k=0j[γ(τ[k](z))]+∑k=0j[g(φ(0))χ(τ[k](z))∏m=0k-1[γ(τ[m](z))]]λj-k.
Hence
(2.14)g(z)=g(τ[j+1](z))∏k=0j[γ(τ[k](z))λ]+∑k=0j[g(φ(0))χ(τ[k](z))∏m=0k-1[γ(τ[m](z))]]1λk+1.
Now if w0 is the Denjoy-Wolff point of τ, it suffices to show that
(2.15)|γ(w0)λ|<1.
Suppose the above inequality holds. Then we conclude that there is 0<β<1 and N∈ℕ such that for k>N we have |γ(τ[k](z))/λ|<β<1. Now we break the proof into two parts.
(1) The Denjoy-Wolff point w0 of τ lies inside D, then g(τ[j](z)) converges to g(w0). Hence
(2.16)limj→∞|g(τ[j+1](z))∏k=0j[γ(τ[k](z))λ]|≤limj→∞g(τ[j+1](z))βj-N|∏k=0N[γ(τ[k](z))λ]|=0.
(2) The Denjoy-Wolff point w0 of τ lies on ∂D, then by [31, Lemma  5.1] τ must be parabolic and by [6, Lemma  3.3] there is a constant C such that
(2.17)11-|τ[j](z)|≤Cj.
Thus it follows that
(2.18)|g(τ[j](z))|=|〈g,Kτ[j](z)〉|≤∥g∥·∥Kτ[j](z)∥=∥g∥·11-|τ[j](z)|2≤∥g∥·jC.
Hence
(2.19)limj→∞|g(τ[j+1](z))∏k=0j[γ(τ[k](z))λ]|≤limj→∞g(τ[j+1](z))βj-N|∏k=0N[γ(τ[k](z))λ]|≤limj→∞∥g∥·(j+1)C·βj-N|∏k=0N[γ(τ[k](z))λ]|=0.
Now we show that |γ(w0)/λ|<1. Since a=c and b=-d, we see that
(2.20)|γ(w0)λ|=|(1-2b¯w0)(-b(1-b¯w0)+a¯aw0)λ(w0)(1-b¯w0)2|.
By [30], we have
(2.21)w0=1-|a|2+|b|2-(1-|a|2+|b|2)2-4|b|22b¯.
Applying the assumptions|b|2≥1/2 and 2|a|2+|b|2≤2/3, an easy computation shows that
(2.22)0≤2b¯w0-1≤1-b¯w0.
Also by using Proposition 2.2, 1/λ<(1-|b|2)/|b|2, and by Lemma 2.4, there is αn≥1 such that τ[n](0)=αnb. Therefore
(2.23)|γ(w0)λ|=|(1-2b¯w0)(-b(1-b¯w0)+a¯aw0)λw0(1-b¯w0)2|=(2b¯w0-1)|-b(1-b¯w0)+a¯aw0|λ|w0|(1-b¯w0)2=(2b¯w0-1)|-b(1-b¯limn→∞αnb)+a¯alimn→∞αnb|λ|w0|(1-b¯w0)(1-b¯limn→∞αnb)≤(2b¯w0-1)(limn→∞|b|(1-αn(|b|2+|a|2)))λ|b|(1-b¯w0)(limn→∞1-αn|b|2)<(1-|b|2)(2b¯w0-1)(limn→∞(1-αn(|b|2+|a|2)))|b|2(1-b¯w0)(limn→∞1-αn|b|2)≤1-|b|2|b|2≤1.Proposition 2.6.
Leta=c, b=-d, |b|2≥1/2, and 2|a|2+|b|2≤2/3. Then λ=∥Cψ,φ∥2 satisfies the equation
(2.24)1=∑k=0∞[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1λk+1.Proof.
Since for every integerj≥0, τ[k](φ(0))≠0, in Proposition 2.5 we set z=φ(0), then we have
(2.25)g(φ(0))=∑k=0∞[g(φ(0))χ(τ[k](φ(0)))∏m=0k-1[γ(τ[m](φ(0)))]]1λk+1.
Since φ(0)=τ(0), we see that
(2.26)g(φ(0))=∑k=0∞[g(φ(0))χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1λk+1.
But g(φ(0))≠0, because otherwise Proposition 2.5 would dictate that the function g(z) is identically 0. Thus eigenfunction g must have the property that g(φ(0))≠0. Hence we have
(2.27)1=∑k=0∞[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1λk+1.We define(2.28)F(z)=∑k=0∞[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]zk+1.
Now we characterize the properties of F and by using these properties we obtain a formula for the norm of Cψ,φ. The idea behind Proposition 2.7 is similar to the one found in [30].Proposition 2.7.
Leta=c, b=-d, |b|2≥1/2, and 2|a|2+|b|2≤2/3. Then F(z) has the following properties.(a)
The power series that definesF(z) has radius of convergence r0 larger than 1/λ.(b)
F(x) is non-negative real number for all x in the interval [0,r0).(c)
F′(x)>0 for all x in the interval (0,r0).Proof.
(a)  By Lemma2.4, for each positive integer n there is αn≥1 such that τ[n](0)=αnb, then χ(τ[m+1](0))=1/αm+1≤1. Also in the proof of Proposition 2.5 we have |γ(w0)/λ|<1, hence there is 0<β<1 and N∈ℕ such that if n>N, then
(2.29)|γ(τ[n](0))λ|<β<1.
Now let β<β1<1 and 0<ϵ<λ(β1-β)/β1. Then if n>N we have
(2.30)|γ(τ[n](0))λ|<|γ(τ[n](0))λ-ϵ|<β1.
Therefore there is a constant C such that
(2.31)|∑k=0∞[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1(λ-ϵ)k+1|≤∑k=0∞1λ-ϵ∏m=0k-1|γ(τ[m+1](0))λ-ϵ|≤C∑k=0∞β1k<∞.
By Lemma 2.4, there is strictly increasing sequence αn≥1 such that τ[n](0)=αnb, and by hypothesis |b|>2/2, hence 1-2αn|b|2<1-2|b|2<0. Also we have |a|2+|b|2≤|b|≤|b/w0|<1/αn, so we conclude that -(1-αn|b|2)+|a|2αn<0. Therefore
(2.32)γ(τ[m+1])(0)=γ(αm+1b)=(1-2αm+1|b|2)(-b(1-αm+1|b|2)+|a|2αm+1b)αm+1b(1-αm+1|b|2)2=(1-2αm+1|b|2)(-(1-αm+1|b|2)+|a|2αm+1)αm+1(1-αm+1|b|2)2>0.
Also it is obvious that
(2.33)χ(τ[m+1](0))=-c¯da¯αm+1b=1αm+1>0.
Hence the proof of part (b) is complete.
(c) Every coefficient ofF is positive and so F′(x)>0 for all x in the interval (0,r0).Now we find an equation that involves the norm ofCψ,φ.Theorem 2.8.
Leta=c, b=-d, |b|2≥1/2 and 2|a|2+|b|2≤2/3. Then λ=∥Cψ,φ∥2 is the unique positive real solution of the equation
(2.34)1=∑k=0∞[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1λk+1.Proof.
By Propositions2.6 and 2.7, there is exactly one positive real number λ which satisfies equation (2.34), and this number must be equal to ∥Cψ,φ∥2.Corollary 2.9.
In Theorem2.8 if one replaces a0 with a and b0 with b such that |a|=|a0|, and |b|=|b0|, then norm of Cψ,φ does not change.Proof.
We haveτ[n](0)=αnb. But by Lemma 2.4, αn=1+αn-1|a|2/(1-αn-1|b|2). Hence if one replaces a0 with a and b0 with b such that |a|=|a0| and |b|=|b0|, then αn, γ(τ[m+1](0)) and χ(τ[m+1](0))=1/αm+1 do not change. Hence by (2.34), the norm of Cψ,φ does not change.Example 2.10.
Letφ(z)=az+b and ψ(z)=az-b, where |a|=1/10 and |b|=8/10. Then we have
(2.35)χ(z)=45z,τ(z)=63z-8080z-100,γ(z)=(5-8z)(-16+13z)z(10-8z)2.
For positive integer k0, let λk0 denote the positive solution of
(2.36)1=∑k=0k0[χ(τ[k+1](0))∏m=0k-1[γ(τ[m+1](0))]]1λk+1.
Now by using numerical methods, we have
(2.37)λ10≈1.796745850919,λ20≈1.797084678603,λ30≈1.797084948747,λ50≈1.797084948963,λ70≈1.797084948963,λ100≈1.797084948963.
Hence we see that ∥Cψ,φ∥2≈1.797084948.The hypotheses of Theorem2.8 restrict us to considering the norms of compact operators. In the remainder of this section we investigate the norm and essential norm of a class of noncompact weighted composition operators.Theorem 2.11.
Letφ(z)=azn+b, for some n∈ℕ, where |a|+|b|=1, ψ∈H∞,let α be one of the nth roots of b|a|/a|b| such that ψ has radial limit at α, and let |ψ| attains its supremum on D∪{α} at α. Then
(2.38)1n|a||ψ(α)|≤∥Cψ,φ∥e≤∥Cψ,φ∥≤1|a||ψ(α)|.Proof.
Let0<r<1. Taking β=rα, by a similar proof for unweighted composition operators [28, Proposition  3.13], we have
(2.39)∥Cψ,φ∥e2≥limr→1-∥Cψ,φ*Kβ∥2∥Kβ∥2=limr→1-|ψ(β)|2·limr→1-∥Kφ(β)∥2∥Kβ∥2=|ψ(α)|2·limr→1-1-r21-(rn|a|+|b|)2=1n|a|(|a|+|b|)|ψ(α)|2=1n|a||ψ(α)|2.
Therefore
(2.40)∥Cψ,φ∥e≥1n|a||ψ(α)|.
On the other hand, by [3], we have
(2.41)∥Cψ,φ∥e≤∥Cψ,φ∥≤∥Mψ∥∥Cφ∥≤∥ψ∥∞∥Caz+b∥=1|a||ψ(α)|.
Therefore
(2.42)1n|a||ψ(α)|≤∥Cψ,φ∥e≤∥Cψ,φ∥≤1|a||ψ(α)|.Corollary 2.12.
In Theorem2.11 if n=1, then
(2.43)∥Cψ,φ∥=∥Cψ,φ∥e=1|a||ψ(α)|.Example 2.13.
(1)  Ifφ(z)=(1/2)z+1/2 and ψ(z)=(z+1)/2, then ∥Cψ,φ∥=2.
(2)   Ifφ(z)=(1/3)z+(2/3)i and ψ(z)=z5-2z3+i, then ∥Cψ,φ∥=43.
(3)   Ifφ(z)=-(1/4)iz+3/4 and ψ(z)=(7z5-5z3+2i)/(z2+2), then ∥Cψ,φ∥=28.



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