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

Abstract

We obtain a representation for the norm of certain compact weighted composition operator 𝐶_𝜓,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.

1. Introduction

Let 𝐷 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 𝐾_𝛼(𝑧)=1/(1−𝛼𝑧). An easy computation shows that ⟨𝑓,𝐾_𝛼⟩=𝑓(𝛼) whenever 𝑓∈𝐻². For any analytic self-map 𝜑 of 𝐷, the composition operator 𝐶_𝜑 on 𝐻² is defined by the rule 𝐶_𝜑(𝑓)=𝑓∘𝜑. Every composition operator is bounded, with

11−|𝜑(0)|²≤‖𝐶_𝜑∶𝐻²→𝐻²‖≤1+|𝜑(0)|1−|𝜑(0)| (1.1) (see [1]). We see from expression (1.1) that ‖𝐶_𝜑‖=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 𝐶_𝜑 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 𝐶_𝜑 for certain linear fractional maps 𝜑. In [6], Bourdon et al. determined the norm of 𝐶_𝜑 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 𝐶_𝜑 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].

If 𝜓 is a bounded analytic function on 𝐷 and 𝜑 is an analytic map from 𝐷 into itself, the weighted composition operator 𝐶_𝜓,𝜑 is defined by 𝐶_𝜓,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 𝐶_𝜓,𝜑=𝑀_𝜓𝐶_𝜑, where 𝑀_𝜓 is a multiplication operator and 𝐶_𝜑 is a composition operator. Recall that if 𝜑 is an analytic self-map of 𝐷, then the composition operator 𝐶_𝜑 on 𝐻² is bounded, hence in this case 𝐶_𝜓,𝜑 is bounded, but in general every weighted composition operator 𝐶_𝜓,𝜑 on 𝐻² is not bounded. If 𝐶_𝜓,𝜑 is bounded, then 𝐶_𝜓,𝜑(1)=𝜓 belongs to 𝐻². These operators come up naturally. In 1964, Forelli [15] 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 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 𝐶_𝜓,𝜑 on 𝐻².

2. Norm Calculation

In this section we obtain a representation for the norm of a class of compact weighted composition operators 𝐶_𝜓,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 𝐶_𝜓,𝜑 on 𝐻² when 𝜑(𝑧)=𝑎𝑧^𝑛+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. Let 𝐾_𝑤 be the reproducing kernel at 𝑤. Then 𝐶^∗_𝜓,𝜑𝐾_𝑤=𝜓(𝑤)𝐾_𝜑(𝑤). (2.1) In the next proposition we generalize the lower bound in (1.1).

Proposition 2.2. Let 𝜑 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 ‖𝐶_𝜓,𝜑𝜓‖≥|^(𝑛)(0)|1𝑛!1−|𝜑(0)|². (2.2)

Proof. We note that if 𝑓 is in 𝐻², then for every 𝑛∈ℕ∪{0} we have |𝑓^(𝑛)(0)/𝑛!|≤‖𝑓‖₂. Hence we have ‖𝐶_𝜓,𝜑‖≥‖𝐶_𝜓,𝜑𝐾_𝜑(0)‖‖𝐾_𝜑(0)‖=‖𝜓.(𝐾_𝜑(0)∘𝜑)‖‖𝐾_𝜑(0)‖≥|(𝜓^(𝑛)(0)/𝑛!)(𝐾_𝜑(0)∘𝜑)(0)|‖𝐾_𝜑(0)‖𝜓=|^(𝑛)(0)|1𝑛!1−|𝜑(0)|². (2.3)

Let 𝑇 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 𝐶_𝜓,𝜑 is compact (see the proof of Proposition 2.5). Hence 0=‖𝐶_𝜓,𝜑‖_𝑒<‖𝐶_𝜓,𝜑‖ and so 𝐶_𝜓,𝜑 is norm-attaining.

Now our goal is to find a functional equation that relates an eigenvalue of 𝐶^∗_𝜓,𝜑𝐶_𝜓,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 𝐶_𝜓,𝜑.

Let 𝜑 be an analytic self-map of 𝐷 and let 𝜓 be a bounded analytic map on 𝐷. Then (𝐶_𝜓,𝜑)^∗=(𝑀_𝜓𝐶_𝜑)^∗=𝐶^∗_𝜑𝑀^∗_𝜓=𝐶^∗_𝜑𝑇^∗_𝜓. (2.4)

But if 𝜑(𝑧)=𝑎𝑧+𝑏 such that |𝑎|+|𝑏|≤1, then by [3] or [28] (𝐶_𝜓,𝜑)^∗=𝑇_𝑔𝐶_𝜎𝑇^∗_ℎ𝑇^∗_𝜓=𝑇_𝑔𝐶_𝜎(𝑇_𝜓ℎ)^∗, (2.5)

where ℎ(𝑧)=1, 𝑔(𝑧)=1/−𝑏𝑧+1, and 𝜎(𝑧)=𝑎𝑧/−𝑏𝑧+1.

From now on, unless otherwise stated, we assume that 𝜓(𝑧)=𝑐𝑧+𝑑,𝜑(𝑧)=𝑎𝑧+𝑏, and |𝑎|+|𝑏|≤1. Since 𝑇^∗_𝑧 is the backward shift on 𝐻², we see that 𝐶^∗_𝜓,𝜑𝐶_𝜓,𝜑𝑓(𝑧)=𝑇_𝑔𝐶_𝜎𝑇^∗_𝜓𝑇_𝜓𝐶_𝜑𝑓(𝑧)=𝑇_𝑔𝐶_𝜎𝑇^∗_{𝑐𝑧+𝑑}(𝜓⋅𝑓(𝜑(𝑧)))=𝑇_𝑔𝐶_𝜎(𝑐(𝜓⋅𝑓(𝜑(𝑧))−𝜓⋅𝑓(𝜑(0))𝑧))+𝑇_𝑔𝐶_𝜎(𝑑𝜓(𝑧)⋅𝑓(𝜑(𝑧)))=𝑇_𝑔(𝑐(𝜓(𝜎(𝑧))⋅𝑓(𝜑(𝜎(𝑧)))−𝜓(0)⋅𝑓(𝜑(0))+𝜎(𝑧)))𝑑𝑔(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)))𝑑𝑔(𝑧)𝜓(𝜎(𝑧))⋅𝑓(𝜑(𝜎(𝑧)))=𝛾(𝑧)𝑓(𝜏(𝑧))+𝜒(𝑧)𝑓(𝜑(0)) (2.6) for all 𝑧 in 𝐷 not equal to 0, where (𝛾(𝑧)=𝑐(1−𝑏𝑧)+𝑑𝑎𝑧)(𝑑(1−𝑏𝑧)+𝑎𝑐𝑧)(𝑎𝑧)(1−𝑏𝑧)²,𝜏(𝑧)=(|𝑎|²−|𝑏|²)𝑧+𝑏−−𝑏𝑧+1,𝜒(𝑧)=𝑐𝑑.𝑎𝑧 (2.7)

In particular, if 𝑔 is an eigenfunction for 𝐶^∗_𝜓,𝜑𝐶_𝜓,𝜑 corresponding to an eigenvalue 𝜆, then 𝜆𝑔(𝑧)=𝛾(𝑧)𝑔(𝜏(𝑧))+𝜒(𝑧)𝑔(𝜑(0)). (2.8) 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 ‖𝐶^∗_𝜓,𝜑𝐶_𝜓,𝜑‖. In the trivial case 𝑎=0 we have ‖𝐶_𝜓,𝜑‖=‖𝜓‖₂(1/1−|𝑏|²). Also if 𝑑=0, then ‖𝐶_𝜓,𝜑‖=|𝑐|‖𝐶_𝜑‖ and if 𝑐=0, then ‖𝐶_𝜓,𝜑‖=|𝑑|‖𝐶_𝜑‖. Therefore we assume that 𝑎,𝑏,𝑐,𝑑 are nonzero.

Throughout this paper, we write 𝜏^[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. Let 𝑔 be an eigenfunction for 𝐶^∗_𝜓,𝜑𝐶_𝜓,𝜑 corresponding to an eigenvalue 𝜆, 𝑧∈𝐷 and for each nonnegative integer 𝑗, 𝜏^[𝑗](𝑧)≠0. Then one has 𝜆^𝑗+1𝑔(𝑧)=𝑔(𝜏^[𝑗+1](𝑧))^𝑗_𝑘=0[𝛾(𝜏^[𝑘]+(𝑧))]_𝑗𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘](𝑧))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚](𝑧))]]𝜆^𝑗−𝑘, (2.9) where one takes ∏⁻¹_𝑚=0(⋅)=1.

Lemma 2.4. For each 𝑛∈ℕ, 𝜏^[𝑛](0)=𝛼_𝑛𝑏, where {𝛼_𝑛} is strictly increasing sequence such that 𝛼_𝑛≥1 for each 𝑛∈ℕ. Also 𝛼_𝑛+1=1+𝛼_𝑛|𝑎|²/(1−𝛼_𝑛|𝑏|²).

Proof. (By induction) Since 𝜏(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 𝜏^[𝑛](0)=𝜏(𝜏^[𝑛−1](0))=𝜏(𝛼_𝑛−1𝛼𝑏)=(1+^𝑛−1|𝑎|²1−𝛼_𝑛−1|𝑏|²)𝑏. (2.10) Now if we set 𝛼_𝑛=1+(𝛼_𝑛−1|𝑎|²)/(1−𝛼_𝑛−1|𝑏|²), then 𝜏^[𝑛](0)=𝛼_𝑛𝑏. But by hypothesis 𝛼_𝑛−1<𝛼_𝑛, so 𝛼1+^𝑛−1|𝑎|²1−𝛼_𝑛−1|𝑏|²𝛼<1+^𝑛|𝑎|²1−𝛼_𝑛|𝑏|², (2.11) which implies that 𝛼_𝑛<𝛼_𝑛+1 also 𝜏^[𝑛+1](0)=𝜏(𝛼_𝑛𝑏)=(1+𝛼_𝑛|𝑎|²/(1−𝛼_𝑛|𝑏|²))𝑏. Hence the proof is complete.

Proposition 2.5. Let 𝑎=𝑐, 𝑏=−𝑑 and let 𝜆=‖𝐶_𝜓,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 𝑔(𝑧)=^∞_𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘](𝑧))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚]1(𝑧))]]𝜆^𝑘+1. (2.12)

Proof. Since 2|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 𝐶_𝜑 is compact and, since 𝐶_𝜓,𝜑=𝑀_𝜓𝐶_𝜑, the operator 𝐶_𝜓,𝜑 is compact. Now according to the paragraph after Proposition 2.2, there is function 𝑔 in 𝐻² such that 𝐶^∗_𝜓,𝜑𝐶_𝜓,𝜑𝑔=𝜆𝑔. Let 𝑧∈𝐷 and for each integer 𝑗≥0, 𝜏^[𝑗](𝑧)≠0. By Lemma 2.3, we have 𝜆^𝑗+1𝑔(𝑧)=𝑔(𝜏^[𝑗+1](𝑧))^𝑗_𝑘=0[𝛾(𝜏^[𝑘]+(𝑧))]_𝑗𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘](𝑧))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚](𝑧))]]𝜆^𝑗−𝑘. (2.13) Hence 𝑔(𝑧)=𝑔(𝜏^[𝑗+1](𝑧))^𝑗_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]+^𝑗_𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘](𝑧))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚]1(𝑧))]]𝜆^𝑘+1. (2.14) Now if 𝑤₀ is the Denjoy-Wolff point of 𝜏, it suffices to show that |𝛾(𝑤₀)𝜆|<1. (2.15) 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
lim_𝑗→∞|𝑔(𝜏^[𝑗+1](𝑧))^𝑗_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]|≤lim_𝑗→∞𝑔(𝜏^[𝑗+1](𝑧))𝛽^𝑗−𝑁|^𝑁_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]|=0. (2.16)
(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
11−|𝜏^[𝑗](𝑧)|≤𝐶𝑗. (2.17) Thus it follows that |𝑔(𝜏^[𝑗](𝑧))|=|⟨𝑔,𝐾_{𝜏[𝑗](𝑧)}⟩|≤‖𝑔‖⋅‖𝐾_{𝜏[𝑗](𝑧)}‖=‖𝑔‖⋅11−|𝜏^[𝑗](𝑧)|²≤‖𝑔‖⋅𝑗𝐶. (2.18) Hence lim_𝑗→∞|𝑔(𝜏^[𝑗+1](𝑧))^𝑗_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]|≤lim_𝑗→∞𝑔(𝜏^[𝑗+1](𝑧))𝛽^𝑗−𝑁|^𝑁_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]|≤lim_𝑗→∞‖𝑔‖⋅(𝑗+1)𝐶⋅𝛽^𝑗−𝑁|^𝑁_𝑘=0[𝛾(𝜏^[𝑘](𝑧))𝜆]|=0. (2.19) Now we show that |𝛾(𝑤₀)/𝜆|<1. Since 𝑎=𝑐 and 𝑏=−𝑑, we see that |𝛾(𝑤₀)𝜆|=|(1−2𝑏𝑤₀)(−𝑏(1−𝑏𝑤₀)+𝑎𝑎𝑤₀)𝜆(𝑤₀)(1−𝑏𝑤₀)²|. (2.20) By [30], we have 𝑤₀=1−|𝑎|²+|𝑏|²−(1−|𝑎|²+|𝑏|²)²−4|𝑏|²2𝑏. (2.21) Applying the assumptions|𝑏|²≥1/2 and 2|𝑎|²+|𝑏|²≤2/3, an easy computation shows that 0≤2𝑏𝑤₀−1≤1−𝑏𝑤₀. (2.22) Also by using Proposition 2.2, 1/𝜆<(1−|𝑏|²)/|𝑏|², and by Lemma 2.4, there is 𝛼_𝑛≥1 such that 𝜏^[𝑛](0)=𝛼_𝑛𝑏. Therefore |𝛾(𝑤₀)𝜆|=|(1−2𝑏𝑤₀)(−𝑏(1−𝑏𝑤₀)+𝑎𝑎𝑤₀)𝜆𝑤₀(1−𝑏𝑤₀)²|=(2𝑏𝑤₀−1)|−𝑏(1−𝑏𝑤₀)+𝑎𝑎𝑤₀|𝜆|𝑤₀|(1−𝑏𝑤₀)²=(2𝑏𝑤₀−1)|−𝑏(1−𝑏lim_𝑛→∞𝛼_𝑛𝑏)+𝑎𝑎lim_𝑛→∞𝛼_𝑛𝑏|𝜆|𝑤₀|(1−𝑏𝑤₀)(1−𝑏lim_𝑛→∞𝛼_𝑛≤𝑏)(2𝑏𝑤₀−1)(lim_𝑛→∞|𝑏|(1−𝛼_𝑛(|𝑏|²+|𝑎|²)))𝜆|𝑏|(1−𝑏𝑤₀)(lim_𝑛→∞1−𝛼_𝑛|𝑏|²)<(1−|𝑏|²)(2𝑏𝑤₀−1)(lim_𝑛→∞(1−𝛼_𝑛(|𝑏|²+|𝑎|²)))|𝑏|²(1−𝑏𝑤₀)(lim_𝑛→∞1−𝛼_𝑛|𝑏|²)≤1−|𝑏|²|𝑏|²≤1. (2.23)

Proposition 2.6. Let 𝑎=𝑐, 𝑏=−𝑑, |𝑏|²≥1/2, and 2|𝑎|²+|𝑏|²≤2/3. Then 𝜆=‖𝐶_𝜓,𝜑‖² satisfies the equation 1=^∞_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]]𝜆^𝑘+1. (2.24)

Proof. Since for every integer 𝑗≥0, 𝜏^[𝑘](𝜑(0))≠0, in Proposition 2.5 we set 𝑧=𝜑(0), then we have 𝑔(𝜑(0))=^∞_𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘](𝜑(0)))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚]1(𝜑(0)))]]𝜆^𝑘+1. (2.25) Since 𝜑(0)=𝜏(0), we see that 𝑔(𝜑(0))=^∞_𝑘=0[𝑔(𝜑(0))𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]]𝜆^𝑘+1. (2.26) 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 1=^∞_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]]𝜆^𝑘+1. (2.27)

We define 𝐹(𝑧)=^∞_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1](0))]]𝑧^𝑘+1. (2.28) Now we characterize the properties of 𝐹 and by using these properties we obtain a formula for the norm of 𝐶_𝜓,𝜑. The idea behind Proposition 2.7 is similar to the one found in [30].

Proposition 2.7. Let 𝑎=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 defines 𝐹(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 Lemma 2.4, for each positive integer 𝑛 there is 𝛼_𝑛≥1 such that 𝜏^[𝑛](0)=𝛼_𝑛𝑏, then 𝜒(𝜏^[𝑚+1](0))=1/𝛼_𝑚+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 |𝛾(𝜏^[𝑛](0))𝜆|<𝛽<1. (2.29) Now let 𝛽<𝛽₁<1 and 0<𝜖<𝜆(𝛽₁−𝛽)/𝛽₁. Then if 𝑛>𝑁 we have |𝛾(𝜏^[𝑛](0))𝜆|<|𝛾(𝜏^[𝑛](0))𝜆−𝜖|<𝛽₁. (2.30) Therefore there is a constant 𝐶 such that |^∞_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]](𝜆−𝜖)^𝑘+1|≤^∞_𝑘=01𝜆−𝜖^𝑘−1_𝑚=0|𝛾(𝜏^[𝑚+1](0))|𝜆−𝜖≤𝐶^∞_𝑘=0𝛽^𝑘₁<∞. (2.31) By Lemma 2.4, there is strictly increasing sequence 𝛼_𝑛≥1 such that 𝜏^[𝑛](0)=𝛼_𝑛𝑏, and by hypothesis |𝑏|>√2/2, hence 1−2𝛼_𝑛|𝑏|²<1−2|𝑏|²<0. Also we have |𝑎|²+|𝑏|²≤|𝑏|≤|𝑏/𝑤₀|<1/𝛼_𝑛, so we conclude that −(1−𝛼_𝑛|𝑏|²)+|𝑎|²𝛼_𝑛<0. Therefore
𝛾(𝜏^[𝑚+1])(0)=𝛾(𝛼_𝑚+1=𝑏)(1−2𝛼_𝑚+1|𝑏|²)(−𝑏(1−𝛼_𝑚+1|𝑏|²)+|𝑎|²𝛼_𝑚+1𝑏)𝛼_𝑚+1𝑏(1−𝛼_𝑚+1|𝑏|²)²=(1−2𝛼_𝑚+1|𝑏|²)(−(1−𝛼_𝑚+1|𝑏|²)+|𝑎|²𝛼_𝑚+1)𝛼_𝑚+1(1−𝛼_𝑚+1|𝑏|²)²>0. (2.32) Also it is obvious that 𝜒(𝜏^[𝑚+1]−(0))=𝑐𝑑𝑎𝛼_𝑚+1𝑏=1𝛼_𝑚+1>0. (2.33) Hence the proof of part (b) is complete.
(c) Every coefficient of 𝐹 is positive and so 𝐹^′(𝑥)>0 for all 𝑥 in the interval (0,𝑟₀).

Now we find an equation that involves the norm of 𝐶_𝜓,𝜑.

Theorem 2.8. Let 𝑎=𝑐, 𝑏=−𝑑, |𝑏|²≥1/2 and 2|𝑎|²+|𝑏|²≤2/3. Then 𝜆=‖𝐶_𝜓,𝜑‖² is the unique positive real solution of the equation 1=^∞_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]]𝜆^𝑘+1. (2.34)

Proof. By Propositions 2.6 and 2.7, there is exactly one positive real number 𝜆 which satisfies equation (2.34), and this number must be equal to ‖𝐶_𝜓,𝜑‖².

Corollary 2.9. In Theorem 2.8 if one replaces 𝑎₀ with 𝑎 and 𝑏₀ with 𝑏 such that |𝑎|=|𝑎₀|, and |𝑏|=|𝑏₀|, then norm of 𝐶_𝜓,𝜑 does not change.

Proof. We have 𝜏^[𝑛](0)=𝛼_𝑛𝑏. But by Lemma 2.4, 𝛼_𝑛=1+𝛼_𝑛−1|𝑎|²/(1−𝛼_𝑛−1|𝑏|²). Hence if one replaces 𝑎₀ with 𝑎 and 𝑏₀ with 𝑏 such that |𝑎|=|𝑎₀| and |𝑏|=|𝑏₀|, then 𝛼_𝑛, 𝛾(𝜏^[𝑚+1](0)) and 𝜒(𝜏^[𝑚+1](0))=1/𝛼_𝑚+1 do not change. Hence by (2.34), the norm of 𝐶_𝜓,𝜑 does not change.

Example 2.10. Let 𝜑(𝑧)=𝑎𝑧+𝑏 and 𝜓(𝑧)=𝑎𝑧−𝑏, where |𝑎|=1/10 and |𝑏|=8/10. Then we have 4𝜒(𝑧)=5𝑧,𝜏(𝑧)=63𝑧−8080𝑧−100,𝛾(𝑧)=(5−8𝑧)(−16+13𝑧)𝑧(10−8𝑧)². (2.35) For positive integer 𝑘₀, let 𝜆_𝑘0 denote the positive solution of 1=^𝑘0_𝑘=0[𝜒(𝜏^[𝑘+1](0))^𝑘−1_𝑚=0[𝛾(𝜏^[𝑚+1]1(0))]]𝜆^𝑘+1. (2.36) Now by using numerical methods, we have 𝜆₁₀≈1.796745850919,𝜆₂₀𝜆≈1.797084678603,₃₀≈1.797084948747,𝜆₅₀𝜆≈1.797084948963,₇₀≈1.797084948963,𝜆₁₀₀≈1.797084948963. (2.37) Hence we see that ‖𝐶_𝜓,𝜑‖²≈1.797084948.

The hypotheses of Theorem 2.8 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. Let 𝜑(𝑧)=𝑎𝑧^𝑛+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 1𝑛|𝑎||𝜓(𝛼)|≤‖𝐶_𝜓,𝜑‖_𝑒≤‖𝐶_𝜓,𝜑1‖≤|𝑎||𝜓(𝛼)|. (2.38)

Proof. Let 0<𝑟<1. Taking 𝛽=𝑟𝛼, by a similar proof for unweighted composition operators [28, Proposition  3.13], we have ‖𝐶_𝜓,𝜑‖²_𝑒≥lim_𝑟→1−‖𝐶^∗_𝜓,𝜑𝐾_𝛽‖²‖𝐾_𝛽‖²=lim_𝑟→1−|𝜓(𝛽)|²⋅lim_𝑟→1−‖𝐾_𝜑(𝛽)‖²‖𝐾_𝛽‖²=|𝜓(𝛼)|²⋅lim_𝑟→1−1−𝑟²1−(𝑟^𝑛|𝑎|+|𝑏|)²=1𝑛|𝑎|(|𝑎|+|𝑏|)|𝜓(𝛼)|²=1𝑛|𝑎||𝜓(𝛼)|². (2.39) Therefore ‖𝐶_𝜓,𝜑‖_𝑒≥1𝑛|𝑎||𝜓(𝛼)|. (2.40) On the other hand, by [3], we have ‖𝐶_𝜓,𝜑‖_𝑒≤‖𝐶_𝜓,𝜑‖≤‖𝑀_𝜓‖‖𝐶_𝜑‖≤‖𝜓‖_∞‖𝐶_{𝑎𝑧+𝑏}1‖=|𝑎||𝜓(𝛼)|. (2.41) Therefore 1𝑛|𝑎||𝜓(𝛼)|≤‖𝐶_𝜓,𝜑‖_𝑒≤‖𝐶_𝜓,𝜑1‖≤|𝑎||𝜓(𝛼)|. (2.42)

Corollary 2.12. In Theorem 2.11 if 𝑛=1, then ‖𝐶_𝜓,𝜑‖=‖𝐶_𝜓,𝜑‖_𝑒=1|𝑎||𝜓(𝛼)|. (2.43)

Example 2.13. (1)  If 𝜑(𝑧)=(1/2)𝑧+1/2 and 𝜓(𝑧)=(𝑧+1)/2, then ‖𝐶_𝜓,𝜑‖=√2.
(2)   If 𝜑(𝑧)=(1/3)𝑧+(2/3)𝑖 and 𝜓(𝑧)=𝑧⁵−2𝑧³+𝑖, then ‖𝐶_𝜓,𝜑‖=4√3.
(3)   If 𝜑(𝑧)=−(1/4)𝑖𝑧+3/4 and 𝜓(𝑧)=(7𝑧⁵−5𝑧³+2𝑖)/(𝑧²+2), then ‖𝐶_𝜓,𝜑‖=28.

Acknowledgment

The authors would like to thank the referee for his valuable comments and suggestions.