Compactification, and beyond, of composition operators on Hardy spaces by weights

Annales Fennici Mathematici Volumen 46, 2021, 43–57 COMPACTIFICATION, AND BEYOND, OF COMPOSITION OPERATORS ON HARDY SPACES BY WEIGHTS Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza Université d’Artois, Laboratoire de Mathématiques de Lens (LML) UR 2462 & Fédération Mathématique des Hauts-de-France FR 2037 CNRS, Faculté Jean Perrin Rue Jean Souvraz, S.P. 18, F-62 300 Lens, France; [email protected] Université d’Artois, Laboratoire de Mathématiques de Lens (LML) UR 2462 & Fédération Mathématique des Hauts-de-France FR 2037 CNRS, Faculté Jean Perrin Rue Jean Souvraz, S.P. 18, F-62 300 Lens, France; [email protected] Université Lille Nord de France, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération Mathématique des Hauts-de-France FR 2037 CNRS F-59 655 Villeneuve-d’Ascq Cedex, France; [email protected] Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS Calle Tarfia s/n 41 012 Sevilla, Spain; pia[email protected] Abstract. We study when multiplication by a weight can turn a non-compact composition operator on H2into a compact operator, and when it can be in Schatten classes. The q-summing case in Hpis considered. We also study when this multiplication can turn a compact composition operator into a non-compact one. 1. Introduction Let ϕ:D→Dbe an analytic self-map and Cϕ:H2→H2be the associated composition operator f7→ f◦ϕ. For w∈H2, the multiplication operator Mwis defined formally by f7→ wf and the weighted composition operator by f7→ w(f◦ϕ). It is known (see [5] for instance) that twisting Cϕby some Mwcan improve its compactness properties, and even its membership in Schatten classes Spor the decay of its approximation numbers [7, Theorem 2.3]. In this note, we study, in a rather qualitative way, the following problem: given a symbol ϕ, when can we find a non-trivial w∈H2such that Mwhas a smoothing effect on Cϕ, namely when is MwCϕcompact if Cϕwas not? Or the other way round: when can we find wsuch that MwCϕis not compact if Cϕwas? In [13, Proposition 2.4], it is proved that for MwCϕto be compact for some w∈H2(w6≡ 0), it is necessary that (1.1) m({|ϕ∗|= 1}) = 0, where mis the normalized Lebesgue measure on Tand ϕ∗the boundary values function of ϕ. On the other hand, in order that MwCϕbe Hilbert–Schmidt for some w∈H2,w6≡ 0, it is sufficient that (1.2) ˆT log(1 −|ϕ∗|)dm > −∞ https://doi.org/10.5186/aasfm.2021.4602 2020 Mathematics Subject Classification: Primary 47B33; Secondary 46B28. Key words: Approximation numbers, composition operator, compactification, decompactification, Hilbert–Schmidt operator, p-summing operators, Schatten classes. 44 Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza [13, Proposition 2.5]. Note that (1.1) means that ϕis not an exposed point of the unit ball of H∞[1], and that (1.2) means that it is not an extreme point of this unit ball [4, Theorem 7.9]. There is a gap between these two conditions. The purpose of this work to fill this gap in several respects, this filling explaining in passing the initial gap. In Section 3, we show that condition (1.1) is necessary and sufficient to have a compact weighted composition operator. We also give examples showing how small approximation numbers we can obtain. In Section 4, we show that condition (1.2) is necessary and sufficient to get a Hilbert–Schmidt weighted composition operator, and we show that it is also necessary and sufficient for getting a weighted composition operator in some, or all, Schatten classes. In Section 5, we consider the case of Hpspaces and study the nuclearity and the summing properties of the weighted composition operators. In Section 6 we show that a composition operator can become non-compact by weighting it if and only if the image of the symbol touches the boundary of the unit disk. 2. Notation Let Dbe the open unit disk. The Hardy space Hp,1≤p < ∞, is the space of analytic functions f:D→Csuch that kfkp p:= sup 0<r<1 1 2πˆ2π 0|f(reit)|pdt < ∞. Such functions have non-tangential limits f∗(eit)almost everywhere on T=∂Dand we have kfkp p=1 2πˆ2π 0|f∗(eit)|pdt. For p= 2,H2is equivalently the space of analytic functions in Dthat can be written f(z) = P∞ n=0 cnznwith kfk2 2=P∞ n=0 |cn|2<∞. In the sequel, for convenience, we write simply k.k2=k.k. Any analytic self-map ϕ:D→Dinduces a bounded operator Cϕ:Hp→Hp, called the composition operator of symbol ϕ. For w∈Hp, the multiplication operator Mwis defined, formally, by Mwf=w f, and the weighted composition operator MwCϕby (MwCϕ)(f) = w(f◦ϕ). Note that to get MwCϕ:Hp→Hp, it is necessary to have w∈Hpsince (MwCϕ)(1) = w. Throughout this paper it will be assumed that w∈Hp, and that w6≡ 0. This membership is not sufficient in general; however w∈H∞is sufficient (but not necessary!), since H∞is the set of multipliers of Hp. Note that we may consider the bounded operator MwCϕ, even if Mwis not bounded. Except in Section 5, we work only with the Hilbert space H2. For convenience, we will adopt in this paper the following terminology. Definition 2.1. We say that the symbol ϕis -compactifiable if MwCϕis compact for some w∈H2with w6≡ 0; -decompactifiable if MwCϕis bounded but not compact for some w∈H2. For ξ∈T=∂Dand 0< h < 1, the Carleson window W(ξ, h)is defined as (2.1) W(ξ, h) = {z∈D; 1 −h≤ |z|and |arg(zξ)| ≤ πh}. Compactification, and beyond, of composition operators on Hardy spaces by weights 45 If µis a positive measure on D, the Carleson function of µis (2.2) ρµ(h) = sup ξ∈T µ[W(ξ, h)]. The measure µis called a Carleson measure when ρµ(h) = O (h), and a vanishing Carleson measure when ρµ(h) = o (h). By the Carleson embedding theorem, this is equivalent to say that the canonical inclusion Jµ:H2→L2(µ)is respectively bounded or compact. It is convenient to coin the Hastings–Luecking box f W(ξ, h)⊆W(ξ, h)defined by (2.3) f W(ξ, h) = {z∈D; 1 −h≤ |z|<1−h/2and −πh < arg(zξ)≤πh}. We denote mthe Haar measure (normalized Lebesgue measure) of T. For a symbol ϕ,mϕ=ϕ∗(m)is the pull-back measure of mby ϕ∗:T→C, the (almost everywhere defined) radial limit function associated with ϕ: (2.4) ϕ∗(ξ) = lim r→1−ϕ(rξ). By definition mϕ(B) = m[ϕ∗−1(B)] for all Borel sets B⊆D. This measure mϕis always a Carleson measure, due to the Littlewood subordination principle. The Carleson function of ϕis that of mϕand is denoted ρϕ: (2.5) ρϕ(h) = sup ξ∈T mϕ∗−1[W(ξ, h)]. When the composition operator Cϕis compact on H2, we have |ϕ∗|<1a.e., and mϕis supported by D. Moreover, mϕis then a vanishing Carleson measure. Recall that a compact operator Tbetween separable Hilbert spaces H1and H2is in the Schatten class Sp=Sp(H1, H2),p > 0, if Pn≥1[sn(T)]p<∞, where sn(T)is the sequence of singular numbers of T, i.e. the eigenvalues, arranged in non-increasing order, of |T|=√T∗T. For p= 2,S2(H1, H2)is the Hilbert–Schmidt class. Let us also recall that, for p≥2, we have T∈Spif and only if PnkT enkp<∞ for every orthonormal basis (en)of H1, and, for p≤2, we have T∈Spif and only if PnkTenkp<∞for some orthonormal basis (en)of H1(see [6] for instance). It follows that if S, T :H1→H2are two compact operators such that kSxk ≤ kTxkfor all x∈H1, then, for all p > 0,T∈Spimplies S∈Sp. We recall Luecking’s theorem [14]. Theorem 2.2. (Luecking’s theorem) Let µbe a positive Borel measure on D. Then the canonical inclusion Jµ:H2→L2(µ)is in the Schatten class Sp,p > 0, if and only if ∞ X n=0 2n−1 X j=0 [2nµ(f Wn,j)]p/2<∞, where f Wn,j =f W(e2jiπ/2n,2−n). Let us point out that the above condition can be replaced by the following variant [8, Proposition 3.3]: ∞ X n=0 2n−1 X j=0 [2nµ(Wn,j)]p/2<∞, where Wn,j =W(e2jiπ/2n,2−n). As usual, the notation A.Bmeans that A≤c B for some positive constant c, and A≈Bmeans that A.Band B.A. 46 Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza 3. Compactification Theorem 3.1. An analytic self-map ϕ:D→Dis compactifiable if and only if m({|ϕ∗|= 1}) = 0. Proof. The necessary part is proved in [13, Proposition 2.4]. Let us recall the easy proof of this fact. Indeed, suppose that T=MwCϕis compact and that |ϕ∗|= 1 on E, with m(E)>0. Since (zn)nconverges weakly to 0in H2and since T(zn) = w ϕn, we should have ˆE|w∗|2dm =ˆE|w∗|2|ϕ∗|2ndm ≤ˆT|w∗|2|ϕ∗|2ndm =kT(zn)k2−→ n→∞0 ; but this would imply that wis null a.e. on Eand hence w≡0(see [4], Theorem 2.2), which was excluded. Let us now prove the sufficient condition. Assume that m({|ϕ∗|= 1}) = 0 holds. Given w∈H2, we can write kMwCϕ(f)k2=ˆT|w∗|2|f◦ϕ∗|2dm =ˆD|f|2dν, where ν=νw=ϕ∗(|w∗|2m), that is ν(A) = ´ϕ∗−1(A)|w∗|2dm. By the Carleson embedding theorem (see [2, page 129]), a necessary and sufficient condition for the operator MwCϕ:H2→H2to be compact is that νis a vanishing Carleson measure for H2. We now produce a suitable w,w6≡ 0. Let (3.1) Γh={z; 1 −h≤ |z|<1} and set Fn=ϕ∗−1(Γ2−n)and cn=m(Fn). Our assumption implies that cn−→ n→∞0. We can hence find an increasing sequence (kn)n≥1of integers such that (3.2) ∞ X n=1 cknlog n < ∞. Let φn:T→R+be defined as φn=(1 non Fkn, 1on T\Fkn. Let wnbe the associated outer function, satisfying |w∗ n|=φn, namely wn= exp (−ψn), with ψn(z) = ˆT 1 + ze−it 1−ze−it log 1 φn(t)dm(t) = log nˆFkn 1 + ze−it 1−ze−it dm(t). Observe that Reψn(z) = log n´FknPz(t)dm(t), where Pz(t) = 1−|z|2 |1−ze−it|2is the Poisson kernel, so that Reψn(z)≥0and |wn(z)| ≤ 1. Moreover |w∗ n|=1 non Fkn. The condition (3.2) ensures that the infinite product w=Qnwnconverges uniformly on compact subsets of D, and defines a function w∈H∞, bounded by 1and without zeros. Indeed, since Reψn≥0, we see that |1−wn(z)| ≤ |ψn(z)| ≤ log nˆFkn 1 + |z| 1−|z|dm(t) = (cknlog n)1 + |z| 1−|z|; Compactification, and beyond, of composition operators on Hardy spaces by weights 47 subsequently, the series P(1−wn)converges normally on compact subsets of D, and the infinite product Qwnconverges uniformly on compact subsets of D, as claimed. The weighted composition operator MwCϕis bounded since w∈H∞. Let finally 0< h < 2−k1and n=n(h)such that 2−kn+1 ≤h < 2−kn. Let ξ∈T. Then W(ξ, h)⊆Γh, so that ϕ∗−1[W(ξ, h)] ⊆ϕ∗−1(Γh)⊆ϕ∗−1(Γ2−kn) = Fkn. As a consequence, |w∗(u)| ≤ |w∗ n(u)| ≤ 1 nfor all u∈ϕ∗−1[W(ξ, h)], and ν[W(ξ, h)] = ˆϕ∗−1[W(ξ,h)] |w∗|2dm ≤1 n2mϕ[W(ξ, h)] ≤1 n2Ch, because we know (see [2, page 129]) that mϕis a Carleson measure. This ends the proof, since n=n(h)tends to ∞when hgoes to 0. Remark. The previous argument can be sometimes quantified, and the degree of compactness of MwCϕspecified (even if there are limitations, as shown by the forthcoming Theorem 4.1). Recall that if T:X→Yis an operator between Banach spaces Xand Y, its n-th approximation number an(T)is defined, for n≥1, as an(T) = inf{kT−Rk;R:X→Yhas rank < n}. When Xand Yare Hilbert spaces, we have an(T) = sn(T), the n-th singular number of T. Theorem 3.2. For each γwith 0< γ < 1/2, there exist a non-compact composition operator Cϕ:H2→H2and a weight w∈H∞such that, for some constant b > 0, we have an(MwCϕ).exp(−b nγ). In particular, MwCϕbelongs to all Schatten classes Sp(H2),p > 0. For the proof, we recall the following simple result. Proposition 3.3. Let νbe a vanishing Carleson measure on D. Then an(Jν).inf 0<h<1e−nh + sup 0≤t≤hrρν(t) t, where Jν:H2→L2(ν)is the canonical inclusion. In particular, if w∈H∞and ϕis a symbol, we have an(MwCϕ).inf 0<h<1e−nh + sup 0≤t≤hrρν(t) t, where ν=ϕ∗(|w∗|2m)is the pull-back measure of |w∗|2mby ϕ∗. For the proof of Proposition 3.3, we refer to [12, Theorem 5.1], where the result is given only for composition operators, but working exactly the same for inclusions, except only that we have to replace the quantity pρν(h)/h by sup0≤t≤hpρν(t)/t. For the special case, just use that kJνfk=k(MwCϕ)fkfor all f∈H2, so there exist two contractions U:L2(ν)→H2and V:H2→H2such that (MwCϕ) = UJνand Jν=V(MwCϕ), and hence an(MwCϕ) = an(Jν). 48 Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza Proof of Theorem 3.2. We use a construction made in [8, Section 3.2]. Let 1< β ≤2and (3.3) u(t) = |sin(t/2)|β. There is an analytic function U:D→Π+={Rez > 0}whose boundary values are (3.4) U∗(eit) = u(t) + iHu(t), where His the Hilbert transform. The symbol ϕis defined, for z∈D, as (3.5) ϕ(z) = exp −U(z). By [8, Lemma 3.6 and Lemma 4.3], the composition operator Cϕ:H2→H2is not compact. Moreover, since |ϕ∗(eit)|= exp −|sin(t/2)|β, we have |ϕ∗(eit)| ≥ 1−h⇐⇒ |t| ≤ log 1 1−h1/β ≈h1/β; so, if Γhis the annulus {z; 1 −h≤ |z|<1}, and we set Fk=ϕ∗−1(Γ2−k), we have ck:= m(Fk)≈2−k/β. Now, let δk= exp(−2k/β/k2). We slightly modify the example of Theorem 3.1 as follows: φk=(δkon Fk, 1on T\Fk. Then, the series Pk≥1cklog(1/δk)converges since cklog(1/δk).1/k2. As in the proof of Theorem 3.1, we can define an outer function wsuch that |w∗|=Qk≥1φk. The same computation gives us, for any Carleson window W(ξ, t)and for ν= ϕ∗(|w∗|2m): ν[W(ξ, t)] .δ2 jt, for 2−j−1≤t < 2−j. Let 0< h < 1arbitrary. There exists an integer l≥0such that 2−l−1≤h < 2−l. Then for 0< t ≤h, we have 2−j−1≤t < 2−jfor some j≥l; hence ρν(t) t.δ2 j≤δ2 l. Therefore Proposition 3.3 gives an(MwCϕ).inf l∈N(e−n2−l+δl).inf l≥0exp(−n2−l) + exp(−2l/β/l2). The choice l=hβ (β+1) log 2 log nigives, for some b > 0: an(MwCϕ).exp −b n1/(β+1)/(log n)2. Now, if 0< γ < 1/2, we take βsuch that 1< β < 1 γ−1and β≤2. We obtain, with another b > 0: an(MwCϕ).exp(−b nγ), as claimed.  Remark 1. For β < 1, since we have mϕ(Γh)≈h1/β, the composition operator Cϕis already compact. When β= 1, we have mϕ(Γh)≈h, but it can be checked that nevertheless Cϕis compact and ρϕ(h) = O h/ log(1/h)(see [8, Remark 3, Compactification, and beyond, of composition operators on Hardy spaces by weights 49 page 3117]). Without doing that, we can use [8, Theorem 4.1] (which is an improvement of [9, Theorem 4.1]): there exists a compact composition operator with symbol eϕsuch that |eϕ∗|=|ϕ∗|; therefore meϕ(Γh) = mϕ(Γh)≈h. For β= 1, the above proof only gives an(MwCϕ).exp −b n1/2/(log n)2. Though in this case Cϕwas already compact, that nevertheless allows to improve the compactness. Remark 2. The case β= 2 corresponds to the simple symbol ϕ(z) = 1+z 2. Indeed, we only used in our construction the modulus of the symbol and for this ϕ, we have |ϕ∗(eit)|=|cos(t/2)| ≈ 1−t2/8≈exp −|sin(t/2√2)|2. We get the following result. Theorem 3.4. Let ϕ(z) = 1+z 2. For each decreasing sequence (εk)of positive numbers such that (δk) = (2k/2ε2k)is decreasing, there exist a weight w∈H∞and a positive constant bsuch that an(MwCϕ).exp −b n1/3εn. Proof. We only have to modify the proof of Theorem 3.2: we replace Fkby Fk=ϕ∗−1(Γ4k/3) so ck=m(Fk)≈2−k/3, and we replace δk= exp(−2k/β/k2) = exp(−2k/2/k2)by δk= exp(−2k/3ε2k), where (εk)kis a given decreasing sequence of positive integers such that (δk)is decreasing. Note that, since (δk)is decreasing, we have ε2k.2−k/2, so Pkε2k<∞. We get an(MwCϕ).inf l≥0e−n4−l/3+ e−2l/3ε2l, and, with l=log n/ log 2, we get, since εn≤ε2l, for some b > 0: an(MwCϕ).exp −b n1/3εn. For example, with εk= 1/(log k)2, we get an(MwCϕ).e(−b n1/3/(log n)2). Theorem 3.4 improves a result of [7, Theorem 2.3], where for this symbol and a given α > 0, weights ware obtained such that an(MwCϕ).log n nα . 4. Hilbert–Schmidt and Schatten regularizations We begin with a characterization of the symbols that can give a Hilbert–Schmidt weighted composition operator. Theorem 4.1. An analytic self-map ϕ:D→Dcan induce a Hilbert–Schmidt weighted composition operator MwCϕ, for some weight w∈H2, if and only if ˆT log 1 1−|ϕ∗|dm < +∞. 50 Pascal Lefèvre, Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza Proof. That the condition is sufficient is proved in [13, Proposition 2.5]. For sake of completeness, we recall the argument. The hypothesis implies that there exists an outer function won Dsuch that |w∗|2= 1 −|ϕ∗|. Then, writing T=MwCϕ, we have ∞ X n=0 kT(zn)k2=∞ X n=0 ˆT (1 −|ϕ∗|)|ϕ∗|2ndm =ˆT 1 1 + |ϕ∗|dm < +∞, and Tis Hilbert–Schmidt, as claimed. Let us prove the necessity of the condition. If w∈H2exists such that MwCϕ: H2→H2is Hilbert–Schmidt, we have in particular |ϕ∗|<1m-almost everywhere, by the easy part of Theorem 3.1. Since MwCϕis Hilbert–Schmidt, we have ∞ X n=0 kw ϕnk2=∞ X n=0 k(MwCϕ)(zn)k2<∞, i.e. ˆT|w∗|21 1−|ϕ∗|2dm < ∞. The following lemma, with u=|w∗|2,v= 1 −|ϕ∗|2and α= 1, then shows that ´Tlog 1 1−|ϕ∗|2dm < ∞. In fact, since w∈H2and w6≡ 0, Jensen’s inequality tells that the first condition of that lemma is satisfied.  Lemma 4.2. Let (Ω, ν)be a measure space and u, v : Ω →(0,1] measurable functions such that, for some α > 0: ˆΩ|log u|dν < ∞and ˆΩ uv−αdν < ∞. Then ˆΩ|log v|dν < ∞. Proof. If we set g=v−αand f=uv−α, we have 0≤log g= log f+ log 1 u≤log+f+|log u| ≤ f+|log u|. By hypothesis, f(which is positive) and |log u|are integrable; hence log gis integrable and ˆΩ|log v|dν < ∞. In Theorem 4.1, we showed that for the outer function wsuch that |w∗|2= 1−|ϕ∗|, the weighted composition operator MwCϕis Hilbert–Schmidt. For this weight, we cannot expect better in general, as said by the following theorem. Theorem 4.3. There exists a symbol ϕsatisfying ´Tlog(1−|ϕ∗|)dm > −∞such that, if wis any outer function satisfying |w∗|= 1 −|ϕ∗|, the weighted composition operator MwCϕis Hilbert–Schmidt, but MwCϕ/∈Sp, for all p < 2. Proof. Let, for |t| ≤ π: u(t) = 1 −exp(−e1/|t|). We have 0<1−exp(−e1/π)≤u(t)≤1; hence ´π −πlog u(t)dt > −∞; therefore there is an outer function ϕ∈H∞such that |ϕ∗(eit)|=u(t). Moreover, we also have ´Tlog(1 −|ϕ∗|)dm ≈´π −πlog 1−u(t)dt > −∞. Hence if wis an outer function such that |w∗|2= 1−|ϕ∗|, the weighted composition operator Compactification, and beyond, of composition operators on Hardy spaces by weights 51 MwCϕis Hilbert–Schmidt. We are going to show that MwCϕdoes not belong to any Schatten class for p < 2. For that, we use Theorem 2.2. The weighted composition operator MwCϕcan be viewed as an inclusion Jν:H2→L2(ν), where ν=ϕ∗(|w∗|2m). Here, we also have dν(z) = (1 −|z|)dmϕ(z). Since p < 2, we have 2n−1 X j=0 [2nν(f Wn,j)]p/2≥2n−1 X j=0 2nν(f Wn,j)p/2 = [2nν(e Γ2−n)]p/2, where e Γh={z∈D; 1 −h≤ |z| ≤ 1−h/2}. But ν(e Γ2−n)≈2−nmϕ(e Γ2−n)and mϕ(e Γh)≈1 (log 1/h)(log log 1/h)2· In fact, we have ϕ∗(eit)∈e Γhif and only if h/2≤exp(−e1/|t|)≤h, which is equivalent to 1 log log 2/h ≤ |t| ≤ 1 log log 1/h and 1 log log 1/h −1 log log 2/h ≈1 (log log 1/h)2log 1 + log 2 log 1/h ≈1 (log 1/h)(log log 1/h)2· Hence 2nν(e Γ2−n)&1 n(log n)2 and we obtain +∞ X n=0 2n−1 X j=0 [2nν(f Wn,j)]p/2& +∞ X n=0 1 np/2(log n)p=∞, since p/2<1. Luecking’s theorem tells that MwCϕ/∈Sp. If Theorem 4.3 does not allow to have a better behavior for MwCϕthan Hilbert– Schmidt when wis an outer function such that |w∗|= 1 −|ϕ∗|, an improvement is possible by taking another weight. Theorem 4.4. Assume that the composition operator Cϕcan induce a Hilbert– Schmidt weighted composition operator. Then there exists another weight w∈H2 such that MwCϕ∈Spfor every p < 2. Proof. By Theorem 4.1, we have ´Dlog 1 1−|z|dmϕ(z)<∞. We will use the following version of the du Bois–Reymond lemma. For sake of completeness, we will give a short proof of this lemma, but we postpone it. Lemma 4.5. (du Bois–Reymond lemma) Let µbe a continuous positive measure on [0,1) and fa positive µ-integrable function. Then there exists a positive function gon [0,1) such that g(t)−→ t→1∞and ´1 0f(t)g(t)dµ(t)<∞. So, let g: [0,1) →R+with g(t)−→ t→1∞and ´Dg(|z|) log 1 1−|z|dmϕ(z)<∞. Then there is an outer function wsuch that |w∗|= (1 −|ϕ∗|)g◦|ϕ∗|.