Volterra operators and semigroups in weighted Banach spaces of analytic functions

Collect. Math. (2014) 65:233–249 DOI 10.1007/s13348-013-0092-5 Volterra operators and semigroups in weighted Banach spaces of analytic functions Manuela Basallote ·Manuel D. Contreras · Carmen Hernández-Mancera ·María J. Martín · Pedro J. Paúl Received: 17 January 2013 / Accepted: 16 July 2013 / Published online: 13 September 2013 © Universitat de Barcelona 2013 Abstract We characterize the boundedness, compactness and weak compactness of Volterra operators Vg(f)(z):= z 0f(ζ)g(ζ) dζacting between different weighted spaces of type H∞ vin terms of the symbol function g, for the case when vis a quasi-normal weight, a notion weaker than normality. Then we apply the characterization of compactness to analyze the behavior of semigroups of composition operators on H∞ v. Keywords Integral operator ·Volterra operator ·Cesàro operator ·Boundedness · Compactness ·Semigroups of analytic functions ·Weighted spaces of analytic functions M. Basallote, M. D. Contreras and C. Hernández-Mancera was partially supported by the Ministerio de Ciencia e Innovación, Spain, and the European Union (FEDER) project MTM2009-14694-C02-02, by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”, and by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133). M. J. Martín was partially supported by grant MTM1009-14694-C02-01, Ministerio de Ciencia e Innovación, Spain, and by the Instituto de Matemáticas de la Universidad de Sevilla (IMUS). P. J. Paúl was partially supported by La Consejería de Economía, Innovación y Ciencia de la Junta de Andalucía (research group FQM-133). M. Basallote ·M. D. Contreras ·C. Hernández-Mancera (B)·P. J. Paúl Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Seville, Spain e-mail: [email protected] M. Basallote e-mail: [email protected] M. D. Contreras e-mail: [email protected] URL: http://personal.us.es/contreras P. J. Paúl e-mail: [email protected] M. J. Martín Departamento de Matemáticas (Módulo 17, Edificio de Ciencias), Universidad Autónoma de Madrid, 28049 Madrid, Spain e-mail: [email protected] URL: http://www.uam.es/mariaj.martin 123 234 M. Basallote et al. Mathematics Subject Classification (2010) Primary 47G10; Secondary 30D15 ·30H05 · 30H10 ·30H30 ·45P05 ·46E15 ·47B07 ·47B33 ·47B38 ·47D06 1 Introduction Given a function gin the space H(D)of complex analytic functions in the unit disk, the familiar term in the integration by part formula Vg(f)(z):= z 0 f(ζ)g(ζ) dζ(z∈D)(1) defines a linear operator Vgon H(D)called the Volterra operator with symbol g.Forg(z)=z we have that Vgis the integration operator, and for g(z)=log(1/(1−z)) we obtain the Cesàro operator. The Volterra operator Vgwas introduced by Pommerenke in [31] to study exponentials of BMOA functions; he proved that Vgis bounded on the Hardy space H2if, and only if, gis a BMOA function. This important result has motivated a number of interesting characterizations of the boundedness and compactness of Vgacting between different types of spaces of analytic functions. To mention only a few, Aleman and Siskakis [5] extended this result to the Hardy spaces Hp(1≤p<∞)and proved that Vgis compact on Hpif, and only if, gis in the VMOA class. Analogous results on some general weighted Bergman spaces were given by these authors in [6] and also by Pau and Peláez [29]. Other relevant papers are [1,4,21,24,25,40,41], and [43]. Similar results in higher dimensions have been also given by Stevi´c(see[38] and references therein). The reader is referred to the nice survey about the origins of Volterra operator, its relevance, and connections with other areas of mathematics, written by Aleman [3]. The first objective of this paper is to study the boundedness and compactness of the Volterra operator Vgacting between weighted Banach spaces of analytic functions H∞ vin terms of the symbol gand the involved weights. Let us recall at this point that a weight vis a non-negative continuous function in Dthat depends only on the radius r=|z|and is decreasing. The weighted Banach spaces H∞ vand H0 vare defined by H∞ v:= f∈H(D):fH∞ v:= sup z∈D v(z)|f(z)|<∞(2) and H0 v:= f∈H∞ v:lim |z|→1v(z)|f(z)|=0.(3) These spaces, which are natural spaces in the sense that norm convergence implies uniform convergence on compact subsets of D, are also known in the literature as growth spaces and are a particular case of mixed norm spaces; see Sect. 2for some historical remarks and the properties we shall use in this paper. Closely related to H∞ vare the Bloch-type spaces B∞ v. A function f∈H(D)belongs to B∞ vwhenever f∈H∞ v. Bloch-type spaces are Banach spaces of analytic functions endowed with the norm fB∞ v:= | f(0)|+sup z∈D v(z)|f(z)|.(4) 123 Volterra operators and semigroups in H∞ v235 An analytic function fbelongs to the little Bloch space B0 vif f∈Bvand lim |z|→1v(z)|f(z)|=0. When the weight vis vα(|z|)=(1−|z|2)α,α>0, the spaces H∞ vαand B∞ vαare the standard weighted Hardy spaces and standard Bloch spaces, respectively, and are usually denoted by H∞ αand B∞ α. In particular, B∞ 1=Bis the classical Bloch space and B0 1=B0is the little Bloch space. A key point in our study of the boundedness and compactness of the Volterra operator acting between two spaces of type H∞ vis the relationship between the growth of a function and the growth of its derivative. It was proved by Hardy and Littlewood in 1932 (see [22, Theorem 39] or [20, p. 80]) that, in some cases, the growth of an analytic function in the unit disk Ddetermines, and is determined by, the growth of its derivative. Namely, for all β>0 we have H∞ β=B∞ β+1. This was extended by Lusky [27], who proved that if the weight v is normal (a well-known class of weights introduced by Shields and Williams [33,34]), then H0 v(r)=B0 (1−r2)v(r)and, by duality, H∞ v(r)=B∞ (1−r2)v(r). Inspired by this result, we say that a weight vis quasi-normal if H0 v(r)=B0 (1−r2)v(r). Sections 3and 4are devoted to our first objective: the study the boundedness and compactness of the Volterra operator Vg:H∞ v1→H∞ v2in terms of its symbol gfor the case when v2is a quasi-normal weight. Our main result in these two sections is Theorem 2where we characterize the compactness and weak compactness of Vg. It turns out that this characterization is useful, via the connection between Volterra operators and semigroups of composition operators [10], to study the maximal subspace of a semigroup of composition operators in the weighted spaces H∞ v; this is done in Sect. 5. Recall that a semigroup of analytic functions (ϕt)t≥0mapping Dinto itself generates a semigroup of composition operators on a Banach space of analytic functions Xwhen the composition operators Ct(f(z)) := f(ϕt(z)) form a semigroup of bounded operators in X. This semigroup is said to be strongly continuous if for all f∈X,wehave lim t→0+Ct(f)−fX=0. When the semigroup is not strongly continuous on X, one looks for the maximal closed subspace of X, denoted by [(ϕt), X],onwhich(ϕt)generates a strongly continuous semigroup of composition operators. The existence of such a maximal subspace, as well as analytical descriptions of it, was obtained in [10]. In Theorem 3of Sect. 5we prove that if the functions of the semigroup fix a point in the unit disk, then [(ϕt), H∞ v]always contains the little space H0 vand, when vis quasi-normal, it never coincides with the big space H∞ v;that is, the semigroup of operators is never strongly continuous on H∞ v. Our final result about semigroups, Corollary 5, characterizes when the maximal subspace [(ϕt), H∞ v]coincides with H0 vin terms of the infinitesimal generator of the semigroup. The paper finishes with a section devoted to offer an understanding of the notion of quasinormal weight, analyzing when H0 v⊆B0 (1−r2)v(r)and B0 (1−r2)v(r)⊆H0 vin terms of intrinsic properties of the weight v. 2 Preliminaries: weighted Banach spaces of analytic functions In this section we review some of the properties of the weighted Banach spaces H∞ vand H0 v, defined by (2)and(3) above, that we will use in this paper. To the best of our knowledge, these 123 236 M. Basallote et al. spaces were first studied by Rubel and Shields [32]. A good reference for their properties is [8]. Of course, many of these properties depend on the weight, so we start by recalling different types of weights which are often considered in the literature. The weight is called typical if lim|z|→1v(z)=0. If vis typical, then (H0 v)∗∗ =H∞ vand the polynomials are dense in H0 v.Thecasewhenlimsup |z|→1v(z)>0 is usually excluded because we have that H∞ vis isomorphic to H∞and H0 v={0}. A weight vis said to be analytic if v(z)=1/f(|z|)for some f∈H(D)that takes real values on [0,1)and is such that |f(z)|≤ f(|z|)in the unit disk. Many results on weighted spaces must be formulated in terms of the associated weight v(z):= 1 sup{| f(z)|: f∈H∞ v,fH∞ v≤1}(5) the supremum being, in fact, a maximum. The associated vis also a weight, satisfies v(z)≤ ˜v(z)for all z∈D, and has the key property that if we take ˜vinstead of v, neither the spaces H∞ vand H0 vnor the norm · H∞ vchange. We will use that if vis typical, then vis also typical and we have v(z)=1 sup{| f(z)|: f∈H0 v,fH∞ v≤1}. A weight vis called essential if there exists a constant C≥1 such that v(z)≤v(z)≤Cv(z)for all z∈D. It is well-known that if vis analytic, then v=˜vand, in particular, it is essential; this property allows us to give plenty of examples of essential weights. Example 1 (a) Take fα(z)=(1−z2)−α(0 <α<∞), then the corresponding analytic weights vα(z)=(1−|z|2)αare sometimes called standard weights. (b) Take β>0and f(z)=exp{1/(1−z2)β}to obtain the weights vexp,β (z)=exp{−1/(1− |z|2)β}. (c) Finally, the analytic functions f(z)=[1−log(1−z2)]−γ, with γ<0, produce the essential weights vlog,γ (z)=[1−log(1−|z|2)]γ. As we mentioned in the Introduction, a fundamental tool in our study is the result due to Hardy and Littlewood (see [22, Theorem 39] or [20, p. 80]) on the relationship between the growth of a function and the growth of its derivative in the unit disk D. Namely, if β>0, then the hypotheses f(z)=O1 (1−|z|2)β and f(z)=O1 (1−|z|2)β+1 are equivalent or, in terms of standard weighted Hardy and Bloch spaces, H∞ β=B∞ β+1.This was generalized by Lusky [27]; to state his result we need the next definition. Definition 1 Following Shields and Williams [34], we say that (a) the weight vsatisfies property (U) if there exists a positive number αsuch that the function r→v(r)/(1−r)αis almost increasing; 123 Volterra operators and semigroups in H∞ v237 (b) the weight vsatisfies property (L)if there exists a positive number βsuch that the function r→v(r)/(1−r)βis almost decreasing; (c) the weight vis normal if it satisfies both properties (U) and (L). In [18,Lemma1],Doma´nski and Lindström proved that a weight vsatisfies property (U) if, and only if, infnv(1−2−n−1) v(1−2−n)>0, and that vsatisfies property (L) if, and only if, there exists a natural number ksuch that lim supnv(1−2−n−k) v(1−2−n)<1. Using these equivalences for the weights introduced in Example 1, one can easily deduce that any weight vαis normal; that any weight vlog,γ satisfies property (U) but it is never normal; and that any weight vexp,β satisfies property (L) but it is never normal. Other normal weights are, for instance, vα,log,γ (r):= (1−r2)α[1−log(1−r2)]γ(where α>0and γ<0) and vlog log,γ (r):= min{1,logγ(1−log(r))}where γ>0. There is a technical characterization of weights with property (L) given by Shields and Williams [34, Lemma 2] that will be used three times in what follows; namely that vhas property (L) if, and only if, sup 0<r<1v(r)r 0 ds v(s)(1−s2)<+∞. (We must warn the reader that Shields and Williams use weights ψ(x)defined in the positive real line that they translate into a weight in the unit disk via the change of variable r=(1−1/x)−1; one must perform this change in order to obtain the condition as it is written above.) Lusky’s extension of Hardy and Littlewood’s result mentioned above can be rewritten, using these equivalences, as follows [27, Theorem 3.1]. Theorem A Assume that the weight vhas property (U). Then vhas property (L) if, and only if, H0 v(r)=B0 (1−r2)v(r). In particular, if vis a normal weight, then H0 v(r)=B0 (1−r2)v(r)and, by duality, H∞ v(r)=B∞ (1−r2)v(r). 3 Boundedness of the Volterra operators In this section, we study the boundedness of the Volterra operators Vg:H∞ v1→H∞ v2and Vg:H0 v1→H0 v2. Some of our results in this section extend previous results obtained by Hu [24] for the case when Vgis defined from H∞ vinto itself and vis a normal weight. Lusky’s Theorem A tells us that if vis normal then H∞ v(r)=B∞ (1−r2)v(r)and this equality is an important technical tool in this context due to the following simple observation: Denote by Mgthe multiplication operator defined by Mg(f)(z):= f(z)g(z),thenVg(f)∈H∞ v(r)= B∞ (1−r2)v(r)if, and only if, Mg(f)∈H∞ (1−r2)v(r)and both elements have comparable norms. This motivates our following definition, which will be widely used along the paper. Definition 2 We say that a weight vis quasi-normal if H0 v(r)=B0 (1−r2)v(r). Note that any quasi-normal weight is typical and that for quasi-normal weights one has H∞ v(r)=B∞ (1−r2)v(r). This is a rather ad-hoc technical definition, of course, and it would be nice to have a characterization of quasi-normal weights in terms of the properties of the weight as a function. We shall devote the final section of our paper to this question. The following lemma will be used in the proof of Theorem 1below. Although the arguments are straightforward, we include the proof for the sake of completeness. 123 238 M. Basallote et al. Lemma 1 Let v1and v2be typical weights such that Vg:H0 v1→H0 v2is bounded. Then V∗∗ g=Vgand, therefore, Vg:H∞ v1→H∞ v2is bounded. Proof If Vg:H0 v1→H0 v2is bounded, then V∗ g:(H0 v2)∗→(H0 v1)∗and V∗∗ g:H∞ v1→H∞ v2 are bounded as well. Consider the elements δz∈H0 v2∗defined by δz(f):= f(z). The span of such functions is dense on H0 v2∗and for all f∈H0 v1we have that V∗ g(δz), f=δz,Vg(f)=z 0 f(ζ)g(ζ) dζ. Now, for f∈H∞ v1we have (Vg)∗∗(f), δz=f,V∗ g(δz). Since the functions fr(z)=f(rz)converge to fas r→1 in the weak-∗topology, it follows that fr,x∗→f,x∗for all x∗∈(H0 v1)∗. Hence, f,V∗ g(δz)=limr→1fr,V∗ g(δz)=limr→1Vg(fr), δz=limr→1z 0fr(ζ)g(ζ) dζ. Finally, since fr→funiformly on compact subsets in the unit disk, we obtain lim r→1z 0 fr(ζ)g(ζ) =z 0 f(ζ)g(ζ) dζ=Vg(f), δz and it follows that (Vg)∗∗ =Vg. Remark 1 To simplify the notation, we shall denote w2(r):= v2(r)(1−r2)throughout. Theorem 1 Let v1and v2be two weights such that v2is quasi-normal. Then, the following conditions are equivalent: (a) Vg:H∞ v1→H∞ v2is bounded. (b) supz∈D v2(z) v1(z)(1−|z|2)|g(z)|<∞. If, in addition, v1is a typical weight, then both (a) and (b) are equivalent to (c) Vg:H0 v1→H0 v2is bounded. Proof To see that (a) implies (b), note first that the inclusion operator I:H∞ v2→B∞ w2is bounded because v2is quasi-normal. Since Vg:H∞ v1→H∞ v2is bounded by hypothesis, we obtain that the multiplication operator Mg:H∞ v1→H∞ w2is bounded. This implies, by [15, Proposition 4.1], that sup z∈D v2(z) v1(z)(1−|z|2)|g(z)|<∞. To see that (b) implies (a), we start by proving that Vg:H∞ v1→B∞ w2is bounded: Take f∈H∞ v1,then Vg(f)B∞ w2=sup z∈D v2(z)(1−|z|2)|f(z)||g(z)| =sup z∈D v2(z) v1(z)(1−|z|2)v1(z)|f(z)||g(z)| ≤sup z∈D v2(z) v1(z)(1−|z|2)|g(z)|fH∞ v1 . 123 Volterra operators and semigroups in H∞ v239 Since neither H∞ v1nor the norm · H∞ v1change if we replace v1by v1,wehavethatVg: H∞ v1→B∞ w2is bounded. Finally, use that H∞ v2=B∞ w2because v2is quasi-normal. Assume now that v1is typical. Since v2being quasi-normal is also typical, we can apply Lemma 1to obtain that (c) implies (a). To see that (b) implies (c), take f∈H0 v1. Using that (b) holds, we obtain lim |z|→1w2(|z|)|Vg(f)(z)|= lim |z|→1v2(z)(1−|z|2)|f(z)||g(z)| ≤sup z∈D v2(z) v1(z)(1−|z|2)|g(z)|lim |z|→1v1(z)|f(z)|=0. This finishes the proof of the theorem.  Remark 2 Danikas and Siskakis proved in [16] that the Cesàro operator, that is, Vgfor g(z)=log (1/(1−z)), is bounded from H∞into BMOA. On the other hand, note that for v1≡1 in the unit disk, we obtain from Theorem 1 that Vgis bounded from H∞into the Banach space of analytic functions H∞ vprovided that vis quasi-normal. However, in this case, H∞ vcontains the Bloch space and, therefore, it contains BMOA as well. Thus, for the particular case v1≡1, Theorem 1follows as a simple consequence of [16, Theorem 1]. Since we may replace v2by its associate weight v2without changing neither the weighted space nor its norm, in the case when v1=v2, we obtain the following corollary. The equivalence between (a) and (c) was proved by Hu [24] for normal weights. Corollary 1 Let vbe a quasi-normal weight. Then, the following are equivalent: (a) the Volterra operator Vgis bounded on H∞ v, (b) the Volterra operator Vgis bounded on H0 v, (c) the symbol g belongs to the Bloch space. Our second corollary below says that in the settings of analytic weights with property (U), the boundedness of Volterra operator is equivalent to property (L). Corollary 2 Assume that vis an analytic weight satisfying property (U). Then, the following conditions are equivalent: (a) The weight vsatisfies property (L) (that is, vis normal). (b) The Volterra operator Vgis bounded on H∞ vfor all g ∈B. (c) Vgis bounded on H∞ vfor g(z)=1 2log 1+z 1−z. Proof By Corollary 1, we only need to prove that (c) implies (a). So assume that (c) holds. By the very definition of analytic weight, there is a holomorphic function in the unit disk f such that fH∞ v=1and f(r)v(r)=1forall0<r<1. Thus, Vg≥Vg(f)H∞ v=sup z∈D v(z)z 0 f(ζ) g(ζ) dζ≥sup 0<r<1v(r)r 0 ds v(s)(1−s2). That is, sup0<r<1v(r)r 0 ds v(s)(1−s2)<+∞, hence vhas property (L) by Shields and Williams’s characterization [34, Lemma 2] mentioned in Sect. 2. 123 240 M. Basallote et al. 4 Compactness of Volterra operators Recall that if Xand Yare Banach spaces and T:X→Yis a linear operator, then T is compact if for every bounded sequence {xn}⊂X, the sequence {T(xn)}has a norm convergent subsequence and Tis weakly compact if for every bounded sequence {xn}⊂X, the sequence {T(xn)}has a weakly convergent subsequence. Every compact operator is weakly compact, but the converse is not true in general. In this section we show that both notions of compactness coincide for Volterra operators between different weighted spaces when the second weight is quasi-normal. We will need the following lemma, that can be proved by a standard argument. Lemma 2 Let v1,v 2be weights such that the operator Vg:H∞ v1→H∞ v2is bounded. Then Vgis weakly compact (resp. compact) if, and only if, for any bounded sequence {fn}in H∞ v1 that converges to zero uniformly on compact subsets of the unit disk, we have that {Vg(fn)} converges weakly to zero (resp. converges to zero in the norm topology of H∞ v1). We are now ready to state our main result about Volterra operators. Theorem 2 Letv1,v 2beweightssuchthatv2isquasi-normal.Then,thefollowingconditions are equivalent. (a) Vg:H∞ v1→H∞ v2is compact. (b) Vg:H∞ v1→H∞ v2is weakly compact. (c) lim|z|→1v2(z) v1(z)(1−|z|2)|g(z)|=0. (d) Vg:H∞ v1→H0 v2is bounded. If, in addition, v1is a typical weight, then the above conditions are equivalent to (e) Vg:H0 v1→H0 v2is compact. (f) Vg:H0 v1→H0 v2is weakly compact. Proof Since v2is quasi-normal, we have that H∞ v2=B∞ w2. Hence, as we pointed out above, Vg(f)∈H∞ v2=B∞ w2if, and only if, Mg(f)∈H∞ w2and both elements have comparable norms. Thus the compactness (resp. weak compactness) of Vg:H∞ v1→H∞ v2is equivalent to the compactness (resp. weak compactness) of Mg:H∞ v1→H∞ w2. But on these spaces Mgis weakly compact if, and only if, it is compact (see [15, Theorem 5.2]). Thus, (a) and (b) are equivalent. We prove now that (b) implies (c). Since Vg:H∞ v1→H∞ v2is weakly compact and the inclusion operator I:H∞ v2→Bw2is bounded, we have that the multiplication operator Mg: H∞ v1→H∞ w2is weakly compact. Using [15, Theorem 5.2], we obtain that Mg:H∞ v1→H∞ w2 is, in fact, compact. This implies, by [15, Corollary 4.3], that lim |z|→1 v2(z) v1(z)(1−|z|2)|g(z)|=0. Let us prove now that (c) implies (d). Since H0 v2=B0 w2, the operator Vg:H∞ v1→H0 v2 is bounded if, and only if, the multiplication operator Mg:H∞ v1→H0 w2is bounded. Note that for all f∈H∞ v1, lim |z|→1(1−|z|2)v2(z)|Mg(f)|= lim |z|→1(1−|z|2)v2(z)|f(z)||g(z)| =lim |z|→1 v2(z) v1(z)(1−|z|2)v1(z)|f(z)||g(z)| ≤lim |z|→1 v2(z) v1(z)(1−|z|2)|g(z)||| f(z)||H∞ v1. 123 Volterra operators and semigroups in H∞ v241 Therefore, using (c), we see that Mg(H∞ v1)⊂H0 w2which is equivalent to (d). Let us see now that (d) implies (b). We will make use of the following useful characterization of weak compactness (see [19,p.482]):“Let T :X→Y be a bounded linear operator between two Banach spaces X and Y.Then,T is weakly compact if, and only if, T∗∗(X∗∗)⊂Y.” Thus, if (d) holds, then V∗∗ g:(H∞ v1)∗∗ →(H0 v2)∗∗ is bounded. Since v2is typical, we obtain that (H0 v2)∗∗ =H∞ v2. Hence V∗∗ g:(H∞ v1)∗∗ →H∞ v2is bounded and this implies that Vg:H∞ v1→H∞ v2is weakly compact. Since Vg(H∞ v1)⊂H0 v2,wealsohavetheweak compactness of Vg:H∞ v1→H0 v2. To prove the second group of equivalences, assume that v1is a typical weight. Bearing in mind that an operator is compact if, and only if, so is its bi-adjoint, it follows that (e) implies (a). If (a), hence (d), holds, we get that Vg:H0 v1→H0 v2is bounded. Using again that its bi-adjoint is compact, we obtain (e). Finally, being clear that (e) implies (f), the proof that (f) implies (c) can be done following the same steps as in the proof that (b) implies (c) above but using, in this case, [15, Corollary 4.5] instead.  Corollary 3 Let vbe a quasi-normal weight. Then, the following conditions are equivalent. (a) Vgis compact on H0 v. (b) Vgis weakly compact on H0 v. (c) Vgis compact on H∞ v. (d) Vgis weakly compact on H∞ v. (e) Vg(H∞ v)⊂H0 v. (f) g∈B0. The equivalence between statements (c) and (f) in Corollary 3wasprovedbyHu[24] under the assumption that the weight is normal. Corollary 4 Let vbe a quasi-normal weight. Then, the following conditions are equivalent. (a) Vg:H∞→H∞ vis compact. (b) Vg:H∞→H∞ vis weakly compact. (c) lim|z|→1v(z)(1−|z|2)|g(z)|=0. (d) Vg:H∞→H0 vis bounded. 5 Semigroups of analytic functions A (one-parameter) semigroup of analytic functions (ϕt)is a continuous homomorphism :t→(t)=ϕtfrom the additive semigroup of non-negative real numbers into the composition semigroup of all analytic functions which map Dinto D.Inotherwords,(ϕt) consists of analytic functions on Dwith ϕt(D)⊂Dfor which the following three conditions hold: 1. ϕ0is the identity in D, 2. ϕt+s=ϕt◦ϕs,for all t,s≥0, 3. ϕt(z)→z,as t→0, for all z∈D. Good references for the properties of semigroups listed below are the books by Abate [2] and Shoikhet [35]. It is worth pointing out that (3) can be replaced by uniform convergence 123 248 M. Basallote et al. that vand uare equivalent too. Thus, valso has property (U). We use again Shields and Williams’s characterization [34, Lemma 2] to obtain that vhas property (L) (because vand uare equivalent) and we conclude that vis normal. 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