Composition operators on the algebra of Dirichlet series

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024) 118:158 https://doi.org/10.1007/s13398-024-01646-4 ORIGINAL PAPER Composition operators on the algebra of Dirichlet series Manuel D. Contreras1·Carlos Gómez-Cabello2·Luis Rodríguez-Piazza3 Received: 2 December 2023 / Accepted: 14 July 2024 / Published online: 3 September 2024 © The Author(s) 2024 Abstract The algebra of Dirichlet series A(C+)consists on those Dirichlet series convergent in the right half-plane C+and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols :C+→C+giving rise to bounded composition operators Cin A(C+)and denote this class by GA. We also characterise when the operator Cis compact in A(C+).Asa byproduct, we show that the weak compactness is equivalent to the compactness for C. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions {t}in the class GAand strongly continuous semigroups of composition operators {Tt},Ttf=f◦t, f∈A(C+). We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C+)and the Hardy spaces of Dirichlet series Hp and H∞. Keywords Composition operators ·Algebra of Dirichlet series ·Semigroups of composition operators ·Banach spaces of Dirichlet series Mathematics Subject Classification Primary 30B50 ·30H50 ·37F44 ·47B33 ·47D03 This research was supported in part by Ministerio de Ciencia e Innovación, Spain, and the European Union (FEDER), project PID2022-136320NB-I00, and Junta de Andalucía, FQM133 and FQM104. BCarlos Gómez-Cabello [email protected] Manuel D. Contreras [email protected] Luis Rodríguez-Piazza [email protected] 1Departamento de Matemática Aplicada II and IMUS, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain 2Departamento de Matemática Aplicada II and IMUS, Escuela Politécnica Superior, Universidad de Sevilla, Calle Virgen de África, 7, 41011 Sevilla, Spain 3Departmento de Análisis Matemático and IMUS, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia, s/n, 41012 Sevilla, Spain 123 158 Page 2 of 26 M. D. Contreras et al. Contents 1 Introduction ............................................... 2 2 The algebra of Dirichlet series A(C+)and its composition operators ................. 4 2.1 Dirichlet series ........................................... 4 2.2 The algebra of Dirichlet series ................................... 5 2.3 Composition operators ....................................... 8 3 The compactness matter .........................................11 4 About the closure of Gordon–Hedenmalm classes ...........................14 5 Semigroups of analytic functions ....................................16 6 Semigroups of composition operators in A(C+)............................17 6.1 The Koenigs function of semigroups in GA............................19 6.2 Some final examples ........................................23 References ..................................................25 1 Introduction Since the introduction of the first Hardy spaces of Dirichlet series by Hendemalm, Linqvist andSeipin[25], these spaces have attired an increasing attention. Nonetheless, when it comes to the algebra of Dirichlet series A(C+), this is, the closure of Dirichlet polynomials in the H∞-norm, the literature is significantly lesser. To the best of the authors’ knowledge, this new space of Dirichlet series was first presented in [5], where some of its most essential properties were proven as well as some non-trivial related results. To mention some others works concerning A(C+),see[29], where the validity of Carlson’s identity in A(C+)is studied, and [1], where the authors consider the maximal Bohr’s strip. One of the most recurrent questions that arises when studying Banach spaces of analytic functions is that of the boundedness of composition operators. Namely, given a Banach space Xof analytic functions in a domain and a function :→, we consider the operator Cgiven by Cf=f◦,for f∈X. One of the goals is to give necessary and sufficient conditions on the symbol so that f◦belongs to X,forall f∈X,andthatCdefines a bounded operator from Xinto itself. In the context of Banach spaces of Dirichlet series, the first characterisation of bounded composition operators was given for the Hilbert space H2by Gordon and Hedenmalm in [24]. To do so, they introduced the now known-as Gordon–Hedenmalm class G. Little after, Bayart obtained in [7] results regarding the boundedness of composition operators on the spaces Hpfor 1 ≤p≤+∞. The complete characterisation of the boundedness for p/∈ 2N∪{+∞}remains a challenging open problem. In the same work, Bayart also characterised the boundedness of composition operators in H∞giving rise to a class of symbols that we denote by G∞. We recall both the definitions of Gand G∞right straightaway. As customary, for θ∈R,wedenoteCθ:= {s∈C:Re(s)>θ}and C+=C0. Definition 1.1 Let :C+→C+be an analytic function. 1. We say that belongs to the class G∞if there exist c∈N∪{0}and a convergent Dirichlet series ϕsuch that (s)=cs+ϕ(s). The value cis known as the characteristic of the function . 2. We say that belongs to the Gordon–Hedenmalm class Gif ∈G∞and (C+)⊂C1/2 in case c=0. 123 Semigroups of composition operators... Page 3 of 26 158 In Sect.2, after recalling some basic notions related to Dirichlet series, we describe the symbolsgivingrisetoboundedcompositionoperatorsinthecontextofthealgebraofDirichlet series A(C+). This leads us to the introduction of the family of symbols GA: Definition 1.2 Let :C+→C+. We say that belongs to the class GAif ∈G∞and, for every M>0, is uniformly continuous in AM:= {s∈C+:0<Re((s)) < M}. Then, the following result is established: Theorem 1.3 (See Theorem 2.13)Let :C+→C+be analytic. Then, Cdefines a bounded composition operator in A(C+)if and only if ∈GA. Furthermore, we show that there are non-uniformly continuous symbols in C+belonging to the class GA.WealsoprovethatGAG∞. Section 3is devoted to the description of compact composition operators in the algebra A(C+), as well as the weakly-compact ones (see Theorem 3.3). In fact, we prove that these classesofoperatorscoincide.Moreover,weshowthatthecompactnessinA(C+)isequivalent tonotfixingacopyofc0.ThecompactcompositionoperatorsinH∞weredescribedbyBayart in [7]. It is also worth mentioning that Lefévre proved in [26] that they coincide with the weakly compact composition operators on H∞. We go beyond this equivalence proving that every non-compact composition operators on H∞fixes a copy of ∞(see Theorem 3.4). Throughout this work and in connection with the boundedness of composition operators on Banach spaces of Dirichlet series, we will be treating with several classes of Dirichlet series symbols. In Sect.4, the closure of the three classes G,G∞and GAunder the local uniform convergence in C+is studied (see Theorems 4.2 and 4.5). Once the description of bounded composition operators is settled, in Sect. 6we focus our attentionon thestudy ofsemigroups ofcomposition operatorson thealgebra A(C+).Berkson and Porta’s work [9] initiated the study of such semigrous in the context of the classical Hardy space of the unit disc H2(D). There, they proved that a semigroup of composition operators {Ct}is strongly continuous in H2(D)if and only if the family of functions {t} is a continuous semigroup of holomorphic functions in the unit disc. Thereafter, several works studying these semigroups in other contexts such has the Bergman spaces, the disc algebra, the Bloch space or BMOA arose. See [2,3,6,10,11,21] and references therein. An analogue version of Berkson and Porta’s theorem in the Hardy spaces of Dirichlet series Hp for 1 ≤p<∞wasestablishedin[18]. More specifically, the semigroups {Ct}are strongly continuous if and only if the semigroup {t}of symbols in Gis continuous. Surprisingly, for these continuous semigroups, the convergence on compact subsets of C+happens to be equivalent to the convergence in half-planes Cε[18, Theorem 4.6]. Afterwards, an even more striking result was found in [17, Theorem 1.2]: every continuous semigroup {t}in the Gordon–Hedenmalm class Gconverges to the identity map uniformly in the right half-plane as ttends to 0. A similar result was proven in the context of the unit disc algebra A(D). In Sect. 6, using that a continuous semigroup {t}in the class G∞is of the form t(s)= s+ϕt(s),ϕt∈D,(see[18,Proposition 3.2])the correspondencebetween stronglycontinuous semigroups of composition operators in A(C+)and continuous semigroups of symbols in the class GAis settled in Theorem 6.4. In Sect.6, some properties of the Koenigs functions of the continuous semigroups in the class GAare studied. These are used to provide some interesting examples of continuous semigroups in G∞. For instance, semigroups in the class G∞which are not in the class GA (see Theorem 6.14). 123 158 Page 4 of 26 M. D. Contreras et al. Section 5is devoted to recall some notions related to continuous semigroups of analytic functions and their infinitesimal generators. Moreover, we also state those theorems from [18]and[17] that will be needed to prove the results of the subsequent sections. 2 The algebra of Dirichlet series A(C+)and its composition operators 2.1 Dirichlet series We denote by Dthe space of convergent Dirichlet series. That is, the family of series ϕ(s)= ∞  n=1 ann−s,{an}∈C which converge in a certain half-plane Cθ,θ∈R. If there are only finitely many nonzero coefficients an,wesaythatϕis a Dirichlet polynomial. If there is only one non-zero coefficient,wethenhaveaDirichlet monomial. In the context of Dirichlet series, the analogue to the radii of convergence of the analytic functions are the abscissae of convergence. The most relevant ones are the following: σc(ϕ) =inf Re(s): ∞  n=1 ann−sis convergent; σb(ϕ) =inf σ: ∞  n=1 ann−sis bounded on Cσ; σu(ϕ) =inf σ: ∞  n=1 ann−sis uniformly convergent on Cσ; σa(ϕ) =inf Re(s): ∞  n=1 ann−sis absolutely convergent. In [12], H. Bohr proved that for every Dirichlet series ϕ, it holds that σu(ϕ) =σb(ϕ). Moreover, there exists the following relation between the abscissae defined above σc(ϕ) ≤σb(ϕ) =σu(ϕ) ≤σa(ϕ) ≤σc(ϕ) +1. A proof of these inequalities can be found in either [27, Section 4.2] or [19, Chapter 1]. There are some basic results concerning Dirichlet series that will arise throughout the exposition. We state them straightaway. The reader may find a proof of the first one in [27, Theorem 8.4.1]. The second one, which is a Montel’s-type principle for Dirichlet series, was first established in [7, Lemma 18] (see also [27, Theorem 6.3.1]). Theorem 2.1 Let σ, ν ∈R. Consider ϕ:Cσ→Cνanalytic such that it can be written as a Dirichlet series in a certain half-plane. Then, σu(ϕ) ≤σ. In particular, ϕis bounded in Cσ+εfor all ε>0. An improved version of this theorem can be found in [14, Theorem 1], where the authors show that Cνcan be replaced by C\{α, β},α= β. Corollary 2.2 Let ϕbe a Dirichlet series in the conditions of the prior theorem. Then, ϕis uniformly continuous in every half-plane Cσ+εfor all ε>0. 123 Semigroups of composition operators... Page 5 of 26 158 Theorem 2.3 Let {fn}be a bounded sequence in H∞. Then, there exist both a subsequence {fnk}and a function f ∈H∞, such that fnkconverges uniformly to f on each half-plane Cε, for all ε>0. The following theorem will also be useful in the proof of several results in forthcoming sections. Theorem 2.4 Suppose that :C+→Cis a function such that k−∈H∞for every k ∈N. Then is analytic in C+and there exist c0∈N∪{0},ϕ∈Dsuch that ϕextends to C+and (s)=c0s+ϕ(s)for all s ∈C+. The statement was implicit in Gordon and Hedenmalm’s seminal work [24]. Years later, it was explicitly stated in [8, Lemma 1] (see also [27, Theorem 8.3.1]). 2.2 The algebra of Dirichlet series In the present work we shall focus our attention in the algebra of Dirichlet series A(C+).We recall that the space H∞consists on those Dirichlet series ϕconverging in C+such that ϕ∞=sup s∈C+ |ϕ(s)|<∞. In fact, thanks to Bohr’s theorem, a Dirichlet series convergent in Cθfor some θ>0and having a bounded analytic extension to C+, belongs to the space H∞.Itisalsousefulto introduce the following family of Banach spaces of Dirichlet, namely, H∞(Cε),ε>0. These spaces consist on those Dirichlet series ϕconverging in Cεsuch that ϕH∞(Cε):= sup s∈Cε |ϕ(s)|<∞. Observethatifwetakeε=0 we recover the space H∞from above. Now, let us give the definition the algebra of Dirichlet series A(C+)and present some of its most basic properties. Definition 2.5 We define the algebra of Dirichlet series, denoted by A(C+), as the collection of all Dirichlet series f(s)= ∞  n=1 ann−s convergent in C+and such that they define a uniformly continuous function fon C+. The following theorem contains some basic properties of the algebra A(C+),see[5, Proposition 2.2, Theorem 2.3] for a proof. Theorem 2.6 The algebra of Dirichlet series A(C+)satisfies the following properties: 1. A(C+)is a closed subspace of H∞. 2. A(C+)=span{n−s:n∈N}·∞. Consequently, the algebra A(C+)is a Banach algebra. The proof of Proposition 2.2 in [5] yields the following more general result on the characterisation of the membership to A(C+). We include the proof to recall the ideas used there. 123 158 Page 6 of 26 M. D. Contreras et al. Proposition 2.7 Let ϕbe a uniformly continuous function in C+, analytic there and such that ϕ∈D. Then, ϕ∈A(C+). Proof We want to show that the Dirichlet series defining ϕis in fact bounded in the whole right half-plane and by Bohr’s theorem it also converges there. Since ϕ∈D, there exists ν∈Rsuch that ϕis bounded in the closed half-plane Cνby, say, M>0. We suppose that ν>0, otherwise there is nothing to prove. By hypothesis, there exists δ>0 such that |ϕ(s1)−ϕ(s2)|<1foreverys1,s2∈{s∈C+:Re(s)<ν}such that |s1−s2|<δ.Take k∈Nsuch that 1 k<δ. We consider s∈C+with Re(s)<ν. Then, |ϕ(s)|≤ k−1  j=0ϕs+j kν−ϕs+j+1 kν+|ϕ(s+ν)|≤k+M. Then, the Dirichlet series ϕis bounded in C+and, consequently, it converges there.  Remark 2.8 Notice that we have only used the uniform continuity of ϕin the vertical strip ={s∈C+:Re(s)≤σu(ϕ)}, so we can ask ϕjust to be a somewhere convergent Dirichlet series which is uniformly continuous in the strip . The algebra A(C+)is another example of how Dirichlet series behave differently to the analytic functions in the unit disc D. Indeed, in the unit disc setting, the space of analytic functions which are uniformly continuous in Dcoincides with those analytic functions having a continuous extension to ∂D. This is no longer true in the case of Dirichlet series. Denote by Bthe class of functions in H∞having an extension to ∂C+. Trivially, Bis also a closed Banach subalgebra of H∞containing A(C+).In[26], Lefévre studied composition operators from Binto H∞.WeseenowthatA(C+)B. Proposition 2.9 There exists a Dirichlet series belonging to H∞such that it has a continuous extension to ∂C+but it is not uniformly continuous in C+. Proof We let T:D→C+be the conformal mapping T(z)=1+z 1−z.Definethemapϕ(s)= 1 22−s−1 23−s. We claim that 1 22−it −1 23−it = 1,for all t∈R. Indeed, if there were a t0∈Rsuch that 2−it0−3−it0=2, then,wewouldhavethatt0=2πk/log 2 and t0=π(2l+1)/ log 3,for k,l∈Z.In particular, this would imply that log 3/log 2 is a rational number, which constitutes a contradiction. Therefore, there is no real number tsuch that ϕ(it)=1. Now, by Kronecker’s Theorem, there exists a sequence {tn}of real numbers such that ϕ(itn)→1asn→∞.Let{hn}be a sequence of positive real numbers such that hn→0. Then, we set sn=hn+itn.We claim the existence of a sequence {rn}in C+such that |rn−sn|→0asn→∞and, setting zn=ϕ(sn),wn=ϕ(rn),that i)zn→1,ii) T(zn) T(wn)=eπ. Regarding i), we know that ϕ(itn)→1. By the choice of hn’s, we still have that zn= ϕ(hn+itn)→1. 123 Semigroups of composition operators... Page 7 of 26 158 To prove ii), first notice that since zn→1, we have that |T(zn)|→∞. Then, take sn=sn+εn,where{εn}is a sequence of positive real numbers tending to zero such that for each n 1 22−εn+3−εn≤1−2eπ |T(zn)|. This gives ii)since, clearly, |un|=|ϕ(sn)|≤1/2(2−εn+3−εn).So, |un|≤1−2eπ |T(zn)|. This is, eπ |T(zn)|≤1−|un| 2≤|1−un| |1+un|=1 |T(un)|. Reordering, we eventually get that  T(zn) T(un)≥eπ. If we take the segments γn(α) =αrn+(1−α)sn,0≤α≤1, we can find αnsuch that |T(zn)|/|T(wn)|=eπ,wherewn=ϕ(γn(αn)). Hence, claim ii)follows with rn=γn(αn). Let A={z∈C:e−π/2<|z|<eπ/2}. Then, the holomorphic function f(z)= exp(ilog T(z)) maps Dinto A. Here log stands for the complex logarithm where the argument is taken in (−π/2,π/2). The Dirichlet series F=f◦ϕis clearly convergent in C+and it satisfies the thesis of the Proposition. Indeed, since ϕcan be clearly continuously extended to ∂C+,1 /∈ϕ(∂C+),and fis continuous in D\{1}, we conclude that Fcan be continuously extended to ∂C+. Regarding the failure of the uniform continuity on C+, consider the sequences {sn}and {rn}from above. By construction, |sn−rn|→0asn→∞. Now, thanks to part ii)of the claim, we have that |Re(log(T(zn))) −Re(log(T(wn)))|=π. Hence, |Im(ilog(T(zn))) −Im(ilog(T(wn)))|=π. Recalling the definition of f,wefindthat |Arg(f(zn)) −Arg(f(wn))|=π. Hence, f(zn)and f(wn)lie in antipodal segments joining the inner circle and the outer circle of the boundary of A. Therefore, between these two points we can fit, at least, a ball of radius e−π/2. Therefore, |f(zn)−f(wn)|≥2e−π/2, and the conclusion follows.  Remark 2.10 The algebra Bis not separable. We can see this using the ideas from the proof of the latter proposition. Let M=2−it :t=(2k+1)π log 3 −log 2,k∈Z. For τ∈∂D,define fτ(z)=exp(ilog T(τ z)),z∈D. Then, we consider Fτ=fτ◦ϕ,where ϕ(s)=1 22−s−1 23−s. We claim the following: i) If τ∈∂D\M,thenFτ∈B. 123 158 Page 8 of 26 M. D. Contreras et al. ii) For τ,τ∈∂D\M,τ= τ: Fτ−Fτ∞≥e−π/2. We begin by showing i). The function fτfails to be continuous whenever τz=1. Hence, so does Fτfor those t∈Rsuch that ϕ(it)=τ.Ifthisoccurs,wehavethat, 1 22−it −1 23−it =τ. This is, the middle-point τof the points 2−it and −3−it in ∂Dlies also in ∂D. This implies that 2−it =−3−it =τ. Or, equivalently, 2it3−it =−1. This forces t=(2k+1)π log 3−log 2 ,k∈Z, and the claim follows. Now, we prove ii). Fix τ,τ∈∂D\M,τ= τ. By Kronecker’s Theorem (see [19, Proposition 3.4]), there exists a sequence {tn}of real numbers such that ϕ(itn)→τas n→∞.Let{hn}be a sequence of positive real numbers such that hn→0. We set sn=hn+itn. Mimicking the construction from the proof of Proposition 2.9,wecanfinda sequence {rn}∈C+,ϕ(rn)→τ, such that |Fτ(sn)−Fτ(rn)|≥2e−π/2.(2.1) Since τ= τ,fτcan be continuously extended to D\{τ}and ϕ(sn), ϕ(rn)→τ,asn→∞, we have that |Fτ(sn)−Fτ(rn)|→0.(2.2) Using both (2.1)and(2.2), 2e−π/2≤|Fτ(sn)−Fτ(rn)|≤2Fτ−Fτ∞+|Fτ(sn)−Fτ(rn)| for all n∈N. Letting n→∞, the claim ii) follows. We have shown that the are uncountable many disjoint open sets in B, giving the nonseparability of the algebra B. 2.3 Composition operators In this section, we give a necessary and sufficient condition for the boundedness of the composition operators C:A(C+)→A(C+). This condition will be given in terms of some properties satisfied by the symbols . In the seminal work [24], Gordon and Hedenmalm characterised the symbols giving rise to bounded composition operators in the Hilbert Dirichlet space H2. To do so, they introduced the now so-called Gordon–Hedenmalm class (see Definition 1.1). Gordon and Hedenmalm proved the following characterisation. Theorem 2.11 (Gordon–Hedenmalm) An analytic function :C1 2→C1 2defines a bounded composition operator C:H2→H2if and only if has a holomorphic extension to C+ that belongs to the class G. The characterisation of the boundedness of composition operators on H∞was given by Bayart in [7](seealso[8, Proposition 2]). Theorem 2.12 (Bayart) A function :C+→C+defines a bounded composition operator C:H∞→H∞if and only if belongs to the class G∞. 123 Semigroups of composition operators... Page 9 of 26 158 The condition belonging to G∞and being uniformly continuous in C+is sufficient to ensure the boundedness of the operator Con A(C+). However, it is no longer necessary (see Example 2.16). The boundedness of Con A(C+)is characterised by the membership of to the class GA, see Definition 1.2, where uniform continuity is only required in the sets AM:= {s∈C+:0<Re((s)) < M},for all M>0. We now prove the main result of the section. Theorem 2.13 Let :C+→C+be analytic. Then, the following statements are equivalent: (a) defines a bounded composition operator Con A(C+). (b) ∈GA. (c) n−∈A(C+)for all n ∈N. (d) ∈G∞and n−∈A(C+)for all n ∈N. (e) ∈G∞and there exists some n ≥2such that n−∈A(C+). Remark 2.14 Notice that in the statement of the theorem we are excluding the case in which the symbol touches the boundary of C+,thisis,(C+)∩∂C+=∅. If this happens, then the symbol is constant and the composition operator Cis trivially bounded on A(C+). Observe also that this is not the case in H∞since not every H∞function is defined in the closure of the right half-plane. Proof of Theorem 2.13 Trivially, d)implies e).Now,sincen−s∈A(C+)for all n∈N,we clearly have that a)implies c). To show c)implies d), it is enough to prove that ∈G∞. To do so, we use Theorem 2.4 and the conclusion immediately follows. Let us show that d)implies a). By Theorem 2.12,Cmaps H∞into itself because ∈G∞. Moreover, we have by hypothesis that Csends the Dirichlet monomials n−s into A(C+). Since the algebra is the closure of the linear span of the monomials in the H∞ norm (Theorem 2.6 (2)) and the operator Cis linear, we conclude that Cactually maps boundedly the algebra of Dirichlet series A(C+)into itself. For b)implies c), consider n∈N,n≥2, since for n=1 there is nothing to prove. We want to show that g(s)=n−(s)is uniformly continuous in C+.Todoso,takeε>0 and M>0sothatn−(M−1)<ε/4. By the uniform continuity of n−sin C+, there exists δ1>0 such that |n−z−n−w|<ε/2 whenever |z−w|<δ 1. By the definition of GA,we have that the symbol is uniformly continuous in AM. Then, there exists a δ>0sothat whenever s1,s2belong to AMand |s1−s2|<δ,wehave|(s1)−(s2)|<δ 1and thus |n−(s1)−n−(s2)|<ε/2. We may assume that δ<δ 1. Now, suppose that neither s1nor s2belong to AM−1. Then, Re((sj)) > M−1, j=1,2. Therefore, by the choice of M>0 |n−(s1)−n−(s2)|≤n−Re((s1)) +n−Re((s2)) ≤2n−(M−1)<ε/2. Eventually, suppose that s1∈AM−1and s2/∈AMand |s1−s2|<δ. By the mean value theorem, if γ(t)=ts1+(1−t)s2,0≤t≤1, there exists t0so that M−1<Re((γ (t0)) < M. Set s3=γ(t0). Clearly, s1,s3∈AMand |s1−s3|<δ. Therefore, |n−(s1)−n−(s3)|<ε/2. In addition, s2,s3/∈AM−1and |n−(s2)−n−(s3)|<ε/2. Hence, |n−(s1)−n−(s2)|≤|n−(s1)−n−(s3)|+|n−(s3)−n−(s2)|<ε 2+ε 2=ε. 123 158 Page 16 of 26 M. D. Contreras et al. Now, by the convergence of the sequence {n}non vertical strips of Cε,wehavethat ϕn→ϕuniformly on those strips. For every n, the Dirichlet series ϕnbelongs to H∞(Cε), ε>0. Thus the sequence {ϕn}nis uniformly Cauchy as ntends to ∞in vertical strips of Cε and, then, ϕnconverges to ϕuniformly in Cε,ε>0 thanks to [22, Section III.5, Exercise 7] (or [28, Chapter 12, Exercise 9]). Consequently, n→uniformly on Cε,ε>0. It remains to study the case when touches the boundary. If this happens, for certain t∈Rwe have (s)=it,foralls∈C+. Regarding the uniform convergence on half-planes, the argument we just carried out also holds for this case.  Remark 4.3 This general fact about the closure of the class G∞∪{iR}under the local uniform convergence can be used to give an alternative proof of [18, Theorem 5.2]. From the very definition of the Gordon–Hedenmalm class Gwe can deduce the following immediate consequence of the theorem we just proved. Corollary 4.4 The class G∪{s∈C:Re(s)=1 2}is closed when equipped with the uniform convergence on compact sets. Nonetheless, the class GAis no longer ‘almost’ closed under the local uniform convergence in C+. The reason being that GA=G∞(see Theorem 4.5 right below) and that there are non-constant symbols in G∞which do not belong to GA(Example 2.15). Theorem 4.5 GA=G∞. Proof First we see that G∞⊂GA.Take∈G∞. By Theorem 4.2 we have that the uniform convergence on compacta is equivalent to the uniform convergence in half-planes Cε,ε>0. Consider the sequence fn(s)=(s+1/n),n∈N. Notice that fn∈GAfor every n. Indeed, is uniformly continuous in every half-plane Cε,ε>0(is bounded in every such half-plane), so any horizontal translation τ,τ>0, is uniformly continuous in C+. Since fn→uniformly in Cε,wehavethat∈GA. The fact that GA⊂G∞follows immediately from the definition of the classes GAand G∞. 5 Semigroups of analytic functions The study of semigroups of analytic functions in the unit disc D(and then in the right halfplane) started in the early 1900s. The work [9], appeared in 1978, due to Berkson and Porta, meant a renewed interest in the study of semigroups of analytic functions. The state of the art can be seen in [13]. We recall the definition straightaway. Definition 5.1 We say that a family {t}t≥0(in short, {t}) of analytic functions t:C+→ C+is a semigroup if the following two algebraic properties are satisfied: •0(s)=s. •For every t,u≥0, t◦u(s)=t+u(s). If, in addition, tconverges to 0uniformly on compact subsets of C+as t→0+,wesay that {t}is a continuous semigroup. One of the goals in this paper is the study of continuous semigroups of analytic functions {t}in the class GAand its connection to the strongly continuous semigroups of composition operatorsin A(C+).See[18]forthestudy ofthisrelationbetweensemigroups ofcomposition 123 Semigroups of composition operators... Page 17 of 26 158 operators in the Hardy spaces of Dirichlet series Hp,p≥1, and semigroups of analytic functions in the classes Gand G∞. In [18, Proposition 3.2], it was proven that for any continuous semigroup in G∞,wehave that t(s)=s+ϕt(s),s∈C+.Thatis,ct=1foreveryt.SinceGA⊂G∞,thesame conclusion holds for continuous semigroups {t}in GA. Notice that this result implies that continuous semigroups of analytic functions in the class G∞coincide with the continuous semigroups in G. If {t}is a semigroup in the class G∞,thenRe(t(s)) ≥Re(s)for all s. This clearly implies that, unless {t}is the trivial semigroup, {t}has no fixed points in C+and, in fact, by [13, Theorem 8.3.1], the Denjoy-Wolff point of the semigroup is ∞,thatis, lim t→+∞ t(s)=∞,s∈C+.(5.1) BerksonandPortaalsoprovedin[9] the existence of the following limit H(s)=lim t→0+ t(s)−s t,for all s∈C+(5.2) and such limit is uniform on compact sets of C+. In particular, His holomorphic. Moreover, t→ t(s)is the solution of the Cauchy problem: ∂t(s) ∂t=H(t(s)) and 0(s)=s∈C+.(5.3) The function His called the infinitesimal generator of the semigroup {t}.In fact, they also proved, see [9, Theorem 2.6], that His the infinitesimal generator of a continuous semigroup of analytic functions with Denjoy-Wolff point ∞if and only if H(C+)⊂C+. Since the Denjoy-Wolff point of a semigroup in the Gordon–Hedenmalm class is ∞ its infinitesimal generator is a holomorphic function sending the right half-plane into its closure. In [18, Theorem 5.1], a description of the infinitesimal generators of the continuous semigroups in G∞was given. Theorem 5.2 Let H :C+→C+be analytic. Then, the following statements are equivalent: (a) H is the infinitesimal generator of a continuous semigroup of elements in the class G. (b) H is bounded in every half-plane Cεand H ∈D. (c) H∈D. Using this description of the infinitesimal generators of continuous semigroups in the class G, the three authors proved in [17] the following result which will play a key role in the proof of the main theorem from Sect.6. Theorem 5.3 Let {t}be a continuous semigroup in the Gordon–Hedenmalm class G,then {t}converges to the identity map uniformly in the right half-plane, as t goes to zero. 6 Semigroups of composition operators in A(C+) There are numerous areas in Analysis where the theory of strongly continuous semigroups of bounded operators on Banach spaces has proven to be quite fruitful. We recall this notion. Definition 6.1 Let Xbe a Banach space and {Tt}t≥0(in short, {Tt}) a family of bounded operators from Xinto itself. We say that {Tt}is a semigroup if the following two algebraic properties are satisfied: 123 158 Page 18 of 26 M. D. Contreras et al. (i) T0=Id, where Id denotes the identity map on X; (ii) For every t,u≥0, Tt◦Tu=Tt+u. If, in addition, it satisfies that (iii) limt→0+Ttf=ffor all f∈X we say that it is a strongly continuous semigroup (also known as C0-semigroup). It is well-known that (iii) is equivalent to the fact that, for each f∈X,themap[0,+∞) t→ Ttfis continuous [20, Page 3, Proposition 1.3]. Clearly, if we have a semigroup of analytic functions {t}, we obtain a semigroup of composition operators {Ct}, whenever such composition operators are well-defined. Conversely, if Tt=Ctis a semigroup of bounded operators in A(C+),{t}is a semigroup in GA. This was established in the setting of Hpspaces, see [18, Proposition 4.3] for a proof of this statement. The same proof holds for the algebra A(C+). The relationship between semigroups of analytic functions in GAand semigroups of composition operators in A(C+)is not only algebraic, but also topological. As in the case of the Hpspaces, for 1 ≤p<∞[18, Theorem 4.6], there exists a one-to-one correspondence between each other. This first result clearly holds in virtue of [18, Theorem 4.4], since any strongly continuous semigroup in A(C+)is strongly continuous in H2. Proposition 6.2 Let {t}be a semigroup in GAand Tt=Ctfor each t. If {Tt}is strongly continuous in A(C+),then{t}is continuous. The following result is a general fact about symbols in GAwhich will be needed in the proof of Theorem 6.4. Proposition 6.3 Let belong to GAwith (s)=cs+ϕ(s).Ifϕ∈H∞,thenis uniformly continuous in C+. Proof Since is in GA, then it is uniformly continuous in the sets AM={s∈C:0<Re((s)) < M},∀M>0. We claim that there exists M0>0 such that 1/2={s∈C:0<Re(s)<1/2}⊂AM0. Indeed, given s∈1/2,wehavethatRe((s)) ≤c/2+ϕ∞=K.TakingM0=K the claim follows. Then, is uniformly continuous in AM0and, in particular, in 1/2.The function is also uniformly continuous in C1/4(Corollary 2.2). Hence, we deduce that  is uniformly continuous in the right half-plane C+. Theorem 6.4 Let {t}be a continuous semigroup of analytic functions in G∞and Tt=Ct, t≥0. Then, the following statements are equivalent: (a) The semigroup {Tt}is strongly continuous in A(C+). (b) t∈GA,t>0. (c) ϕt∈A(C+),t>0. Proof By Theorem 2.13,wehavethata)implies b). By the definition of the class GA, clearly, c)implies b). We now prove the reverse implication. We know that t(s)=s+ϕt(s). By Theorem 5.3, there exists a t0>0 such that ϕt∈H∞for all t∈[0,t0]. By the semigroup structure, we can extend the conclusion to the range t>0. Therefore, every function tis under the hypothesis of Proposition 6.3. Applying this result, we conclude that tis uniformly continuous in C+for every t>0 123 Semigroups of composition operators... Page 19 of 26 158 For the proof of b)implies a)notice that the hypothesis together with Theorem 2.13 imply that Ttf=f◦tbelongs to A(C+)for any fin this space. Theorem 5.3 yields that t(s)→suniformly in C+. Furthermore, every Dirichlet series fin A(C+)is uniformly continuous in C+. Consequently, we conclude that f◦t→funiformly in C+for every f∈A(C+). There exists an analogue version of Theorem 6.4 in the classical setting of the unit disc algebra A(D)(see [16, Theorem 1.2]). Remark 6.5 It is said that {Tt}is uniformly continuous if and only if Ttconverges, as t goes to 0, to the identity map in the norm of the space of linear bounded operators in X. Clearly, every uniformly continuous semigroup is strongly continuous. Using essentially the same argument as in [18, Theorem 5.8] as well as the fact that the multiplication operator f→ n−sfis an isometry in H∞, we can conclude that the only uniformly continuous semigroup of composition operators in A(C+)is the trivial one, namely, Tt=Id for every t. 6.1 The Koenigs function of semigroups in GA The next pages are devoted to the study of some properties of the Koenigs function of continuous semigroups in the class GA. The same thing was done previously for the class Gin [18]. In fact, using some of the properties established there, we are able to construct continuous semigroups of analytic functions in G∞but not in GA. The reader interested in reading about this topic in the more classical setting of continuous semigroups in the unit disc Dis addressed to the book [13, Chapter 9]. Before moving ahead, we provide some definitions that will be needed later. Definition 6.6 Let λ∈Cbe such that Re(λ) ≥0anda domain in Csuch that 0 ∈. (a) We say that is λ-spirallike if e−λt⊂for all t≥0. (b) If λ∈(0,∞),wesaythatis starlike (c) A map h:D→Cis λ-spirallike with respect to τ∈Dif it is univalent, h(τ) =0and h(D)is a λ-spirallike domain. (d) If a map h:D→Cis λ-spirallike with λ∈(0,∞), it is also called starlike. Next Theorem is proven in the unit disc setting in [13, Theorem 9.3.5], but it holds in any simply connected domain . Theorem 6.7 Let {t}be a non-elliptic continuous semigroup of analytic functions in C+. Then, there exists a univalent function h :C+→Csuch that h◦t(s)=h(s)+t.(6.1) The function h is unique up to an additive constant. The function hisknownas theKoenigs function ofthe semigroup. Theinterestabout studying properties of the Koenigs function hlies on the useful information it can provide about the semigroup. In fact, since the function his univalent, we can recover the semigroup {t}as t(s)=h−1(h(s)+t). Now, a derivation with respect to tand evaluating at t=0in(6.1) gives that h(s)=1 H(s), 123 158 Page 20 of 26 M. D. Contreras et al. where His the infinitesimal generator of the continuous semigroup {t}. Therefore, the infinitesimal generator of a continuous semigroup {t}and its Koenigs function are linked via the latter identity. In the setting of continuous semigroups in the class G, there exists a full description of the Koenigs functions of such semigroups given in [18, Corollary 6.8]. We recall it straightaway. Theorem 6.8 Let h be a holomorphic function in C+. Then, h is the Koenigs function of a continuous semigroup in Gif and only if his a Dirichlet series satisfying h(C+)⊂C+\{0}. Take {t}a continuous semigroup in G∞. If for certain b∈Rthe limit lim s→ib t(s) happensto exist,then itis finite.Indeed, thanks toTheorem 5.3,the elementsofthe semigroup are of the form t(s)=s+ϕt(s),whereϕt∈H∞. Therefore, if the limit of tat ∂C+ exists, it is necessarily finite. Theorem 6.9 Let {t}be a continuous semigroup in G∞and h be its Koenigs function. Then, the following assertions are equivalent: (a) For every t >0and for every b ∈R, the limit lims→ib t(s)exists. (b) There exists t0>0such that for every b ∈Rthe limit lims→ib t0(s)exists. (c) For every β∈∂C+∪{∞}, the limit lims→βh(s)exists in the Riemann sphere. Equivalently, h has a continuous extension from C+∪{∞}to the Riemann sphere. (d) tis uniformly continuous on CR={z∈C+:|Im(z)|<R}, for every R >0and t>0. Proof a)implies b)is trivial. For b)implies c), consider L:D→C+the Riemann mapping L(z)=1+z 1−z. Then, define the analytic functions in the unit disc Dgiven by t(s)=L−1◦ t◦L(z),z∈D. Clearly, {t}is a continuous semigroup in D. In fact, its Koenigs function is hD=h◦L,wherehstands for the Koenigs function of the semigroup {t}. Moreover, its Denjoy–Wolff point is 1. Since L:D→C+is a homeomorphism, to prove c)it suffices to check that the limit limz→τhD(z)exists in the Riemann sphere for every τ∈∂D. Thanks to b), the function t0:D→Dhas limit at every point D\{1}.However,t0also has a limit at the point z=1, since t0∈G∞,sot0(s)=s+ϕt0(s), with ϕt0∈H∞. Therefore, lims→∞ t0(s)=∞,andthen, lim z→1t0(z)=1. Then,wehaveproventhatt0belongs to the unit disc algebra A(D). Therefore, by [13, Theorem 11.3.8, c)⇒a)], we conclude that the unrestricted limit of hDexists at every point of ∂D\{1}. By the definition of hD,wehavethathhas a continuous extension to the imaginary axis iR. It remains to prove the existence of its extension to the Denjoy-Wolff point of the semigroup {t}. If the semigroup {t}consists of automorphisms, the result trivially holds. Therefore, assume that they are not automorphisms. Now, by [18, Proposition 6.10], we know that the semigroup {t}has zero-hyperbolic step. Thanks to the isometric nature of the hyperbolic distance for biholomorphisms, see [13, Proposition 1.3.10], we conclude that the semigroup {t}also has zero hyperbolic step. An application of Theorem [13, Theorem 11.1.4, 3) iii)] yields lim z→1hD(z)=∞. 123 Semigroups of composition operators... Page 21 of 26 158 This guarantees that the Koenigs function hDof the semigroup {t}has unrestricted limit at the Denjoy-Wolff point τ=1. Since h=hD◦L−1, we conclude that lim s→∞ h(s)=lim s→∞ hD◦L−1(s)=∞. For c)implies a), we define the semigroup tanalogously and we let hDstand for its Koenigs function. Thanks to c),wehavethathDhas unrestricted limit at D\{1}, and the Denjoy-Wolff of the semigroup is 1. Using [13, Theorem 11.3.8, a)⇒b)], we conclude that tbelongs to the disc algebra A(D)for every t>0. Since, t(s)=L◦t◦L−1(s),wehavethatfor every b∈R, the limit lims→ib t(s)exists. If we assume d)then, for any b∈R, there exists R>0 big enough so that ε+ib, belongs to CRfor every ε>0. Since tis uniformly continuous in CR, we can extend it to CRand, in particular, to the point ib and a)follows. Reciprocally, suppose that a)holds. Take R>0. Then, tcan be extended to CR, by hypothesis. If we restrict ourselves to those points in CRwith real part less or equal than 1/2, we have that, by compactness, t is uniformly continuous there. The uniform continuity of tin every half-plane Cε,ε>0 yields the result.  Remark 6.10 Notice that if {t}is a continuous semigroup in the class G∞such that there exists t0>0 satisfying t0∈GA, then all the conditions from Theorem 6.9 are satisfied. In the next Corollary we show that, in analogy with case of continuous semigroups of holomorphic self maps of the unit disc [13, Theorem 11.3.8], for the case of a certain class of continuous semigroups in G∞, it suffices that an element of the semigroup belongs to GAto conclude that every function of the semigroup is in the class GA. Corollary 6.11 Let {t}be a continuous semigroup in G∞. Assume the existence of t0>0 such that t0∈GAand t1>0such that t1(s)=s+g(k−s), with k ∈N,k≥2. Then, ϕt1∈A(C+). In particular, t1∈GA Proof If there exists t0>0 such that t0is in GA, the conditions from Theorem 6.9 are all satisfied. Hence, we have that for every t,tis uniformly continuous on the closure of CR∩{0<Re(z)<1/2},foreveryR>0 and the same occurs for ϕt.Sinceϕtis periodic (with an imaginary period), we conclude the uniform continuity of ϕtin the whole right half-plane and the claim is proven.  Corollary 6.12 Let {t}be a continuous semigroup in G∞and h its Koenigs function. Assume that ϕt(s)=gt(k−s),g t:D→C+holomorphic, k ∈N,k ≥2and that h is uniformly continuous in C+. Then, t∈GAfor all t. Proof Since his uniformly continuous in C+, by Theorem 6.9,everyt(s)=s+ϕt(s)is uniformly continuous in any horizontal strip of C+of width R>0. Then, by the periodicity of ϕt, every symbol tis uniformly continuous on C+and the claim follows.  In [18, Theorem 5.2], it was shown that given a continuous semigroup {t}in G∞such that t(s)=s+gt(k−s),gt:D→C+holomorphic, then the infinitesimal generator Hof the semigroup is of the form H(s)=G(k−s),whereG:D→C+. Clearly, this also gives that the Koenigs function hof the semigroup is of the form h(s)=F(k−s),whereF:D→C+is holomorphic, too. The reverse implication also holds. More precisely, for {t}a continuous semigroup in the class G∞such that its Koenigs function his given by h(s)=F(k−s), with F:D→C+holomorphic and k∈N,k≥2, the elements tof the semigroup are all of 123 158 Page 22 of 26 M. D. Contreras et al. the form t(s)=s+ϕt(s),whereϕt(s)=gt(k−s),andgt:D→C+is holomorphic for every t. In fact, these specific semigroups can be fully described through spirallike functions on D. Proposition 6.13 Let k ∈N,k≥2. (a) Let f :D→Cbe c-spirallike with respect to the origin. There exists a continuous semigroup {t}in G∞such that its Koenigs function is h(s)=− 1 clog klog(f(k−s)), c>0. Reciprocally, let {t}be a continuous semigroup in G∞such that its Koenigs function h is of the form h(s)=ds +g(k−s),whereg:D\{0}→Cis holomorphic, d ∈C+\{0}. Then, f (z)=zexp(−log(k)g(z)/d)is 1/d-spirallike with d ∈C+\{0}. (b) Moreover, if f cannot be continuously extended to ∂D, then, for every t, t/∈GA. Proof We begin showing the necessity in a).Since fis c-spirallike with respect to zero, f(0)=0. Therefore, the function F(s)=f(k−s)does not vanish in C+and, consequently, it admits a logarithm. Set h(s)=− 1 clogklog(f(k−s)), Re(c)≥0,c= 0. We claim that it is the Koenigs function of a continuous semigroup in G∞. By Theorem 6.8, it suffices to check that his a Dirichlet series such that Re(h)>0. A simple computation shows that h(s)=k−s c f(k−s) f(k−s). Since f:D\{0}→D\{0}and k−ssends the right half-plane into the punctured unit disc, we have that h=G(k−s)with Ga holomorphic function in D. Therefore, his a Dirichlet series. Now, as fis c-spirallike, we conclude that Re(h)>0[13, Theorem 9.4.5] and Theorem 6.8 gives the desired conclusion. For the sufficiency of a),since{t}is a continuous semigroup in G∞,by[18, Corollary 6.8], we have that h(s)=d−log(k)g(k−s)k−sis a Dirichlet series such that Re(h)>0. Taking this into account, a simple computation for those z∈Dsuch that f(z)= 0 yields Re dz f(z) f(z)=Re d−log(k)g(z)z>0, wherewearetakingz=k−s,s∈C+. Regarding part b),if ffails to have a continuous extension to ∂D, then so does the Koenigs function h. An application of Theorem 6.9 yields the desired result.  This description of these ‘periodic’ continuous semigroups in G∞will allow us to provide many examples of continuous semigroups in G∞which fail to be in GA. Theorem 6.14 There exists a continuous semigroup {t}in the class G∞such that tis not in GAfor any t >0. Proof For n∈N, consider the segments γn={z∈D:1/2≤|z|<1,arg(z)=π 2n}and let γ0=[1/2,1).Weset=n≥0γn. Then, the domain =D\is starlike with respect to the origin. We shall now consider the Riemann map ffrom the unit disc Dinto the domain such that f(0)=0. By Proposition 6.13 a), there exists a continuous semigroup {t}in G∞whose Koenigs function is h(s)=− 1 clog klog(f(k−s)), c>0. 123 Semigroups of composition operators... Page 23 of 26 158 Now, by Carathéodory’s Theorem [13, Theorem 4.3.1], since ∂ is not locally connected, f fails to have a continuous extension to ∂D. Then, Proposition 6.13 b)gives the conclusion.  We can also provide examples of a well known fact for semigroups of analytic functions in the unit disc: there exist continuous semigroups {t}in Dsuch that for some t0>0the composition operators Ctare compact in H∞(D)for every t>t0but fail to be compact in H∞(D)for t∈[0,t0]. Proposition 6.15 There exists a continuous semigroup {t}in GAsuch that for some t0>0 the composition operators Ctare compact in A(C+)for t >t0but Ctfails to be compact in A(C+)for every t ∈[0,t0]. Proof Let D1=D\[1/2,1)bethestarlikedomainwithrespecttotheorigin.Let f:D→D1 be the Riemann mapping fixing the origin and such that f(0)>0. In particular, this implies that f((−1,1)) ⊂(−1,1/2). Now, thanks to Proposition 6.13 a), there exists a continuous semigroup {t}in G∞whose Koenigs function is h(s)=− 1 clog 2 log(f(2−s)), c>0. Observe that fcan be continuously extended to D(by Carathéodory’s Theorem or by direct verification). This implies, by Theorem 6.9, that for every t>0, t(s)=s+ϕt(s)is uniformly continuous on horizontal strips of C+(and so is ϕt).Bythe 2πi log 2 -periodicity of ϕt, we conclude that for all t>0, tis uniformly continuous in C+. Hence, {t}∈GA. Now, hmaps C+into D2,whereD2=C+\k∈Z{t+2kπi:t∈[0,1/c]}. Suppose that t>1/c. Then, t(C+)=h−1(h(C+)+t)⊂h−1(Ct). We claim that there exists η>0 such that h−1(Ct)⊂Cη. If this were not the case, there would exists a sequence {sn}∈Ctsuch that Re(sn)→0andRe(h(sn)) > t,foralln.For each n,setzn=2−sn. Then, |zn|→1. Now, |f(zn)|=exp(−clog(2)Re(h(sn))) < exp(−ct log 2)=2−tc <1/2. Nonetheless, since |zn|→1, necessarily f(zn)→∂D1, which constitutes a contradiction with the fact that |f(zn)|<2−tc <1/2. Suppose now that t≤1/c.LetS={z∈C+:0<Im(z)<2πi/log(2)}.Themap φ(s)=f(2−s)is a bijection between the set Sand D\[0,1). Therefore, there exists a sequence {sn}in {z:Im(z)>0}∩Ssuch that f(2−sn)→1. Therefore, considering the principal branch of the logarithm of f(2−sn),wehavethath(sn)→0. Hence, h(sn)+ t→t∈∂D1as n→∞. Then, h−1(h(sn)+t)→∂C+as n→∞. We conclude that Re(t(sn)) →0. Hence, by Theorem 3.3, the semigroup {t}satisfies the desired properties with t0=1/c. 6.2 Some final examples We begin giving a condition on the infinitesimal generator Hso that the corresponding semigroup {t}lies in GA. Lemma 6.16 Let H be a holomorphic function in C+such that H ∈Dand H∈H∞. Then, H∈A(C+). 123 158 Page 24 of 26 M. D. Contreras et al. Proof Since His bounded, His, in fact, Lipschitz-continuous in C+and, consequently, uniformly continuous in C+. Hence, H∈A(C+). Proposition 6.17 Let H :C+→C+be in H∞and such that H∈H∞. Then, the corresponding semigroup {t}tis in GA. Proof The claim follows immediately from Gronwall’s Lemma. Indeed, set M:= H∞, consider s1,s2∈C+and let G(t)=|t(s1)−t(s2)|. Then, for t>0fixed, G(t)=|t(s1)−t(s2)|≤|s1−s2|+t 0 |H(τ(s1)) −H(τ(s2))|dτ ≤G(0)+Mt 0 |τ(s1)−τ(s2)|dτ =G(0)+Mt 0 |G(τ)|dτ. Then, Gronwall’s Lemma yields G(t)≤G(0)exp(Mt). Therefore, for each t>0, the function tis Lipschitz so, in particular, it is uniformly continuous. By Theorem 6.4 the conclusion follows.  Throughout the next examples, {t}will denote a continuous semigroup in the class G∞,H will stand for its infinitesimal generator and hfor its Koenigs function. Example 6.18 {t}∈GAneeds not imply that h ∈A(C+). Leth(s)=1 1−2−s.ByTheorem6.8,its primitive,his theKoenigs functionofacontinuous semigroup {t}in G∞.Now,sinceH=1/his a Dirichlet polynomial, by Proposition 6.17, we conclude that {t}is actually in GA.However,whens=x,x∈(0,ε),ε>0, the function hbehaves like 1/xand its primitive is clearly unbounded close to zero. Example 6.19 {t}∈GAneeds not imply that H ∈H∞. Now, we take h(s)=1+2−s. Its primitive is clearly uniformly continuous in C+.On the other hand, since H=1/h=g(2−s),whereg:D→C+is holomorphic, then the semigroup is of the form t(s)=s+ft(2−s),where ft:D→C+for every t>0. Then, Corollary 6.12 guarantees us that {t}∈GA.However,H(s)=1/(1+2−s)is not in H∞. Example 6.20 H∈H∞needs not imply that {t}∈GA. For n∈N, consider the segments γn={z∈D:1/2≤|z|<1,arg(z)=π 2n} and let γ0=[1/2,1).Weset=n≥0γn. Then, the domain D1=D\is starlike. Consider f:D→D1the Riemann mapping fixing the origin. Proposition 6.13 guarantees the existence of a continuous semigroup {t}in G∞whose Koenigs function is h(s)=− 1 clog 2 log(f(2−s)), c>0. We consider the function h1(s)=h(s)+ds,whereRe(d)>0. Since Re(h)>0and h 1is still a Dirichlet series mapping C+into itself, we conclude that h1is the Koenigs function of a continuous semigroup {t}in G∞. Clearly, |h 1|>Re(h)>Re(d)>0, so H1=1/h1∈H∞.However,since∂ is not locally connected, ffails to have a continuous extension to ∂Dand, consequently, so does h1. Theorem 6.9 prevents the semigroup {t} from having a continuous extension to ∂C+and, therefore, to belong to the class GA. 123 Semigroups of composition operators... Page 25 of 26 158 Acknowledgements We acknowledge Professor J. Bonet for calling our attention to some useful references. Funding Funding for open access publishing: Universidad de Sevilla/CBUA Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Alves, T.R., Brito, L., Carando, D.: Algebras and Banach spaces of Dirichlet series with maximal Bohr’s strip. Rev. Mat. Complut. 36, 607–625 (2023) 2. Anderson, A., Jovovic, M., Smith, W.: Composition semigroups on BMOA and H∞. J. Math. Anal. Appl. 449, 843–852 (2017) 3. Arévalo, I., Contreras, M.D., Rodríguez-Piazza, L.: Semigroups of composition operators and integral operators on mixed norm spaces. Rev. Mat. Complut. 32, 767–798 (2019) 4. Aron, R., Galindo, P., Lindström, M.: Compact homomorphisms between algebras of analytic functions. Stud. Math. 123, 235–247 (1997) 5. Aron, R., Bayart, F., Gauthier, P.M., Maestre, M., Nestoridis, V.: Dirichlet approximation and universal Dirichlet series. Proc. Am. Math. Soc. 145, 4449–4464 (2016) 6. Avicou, C., Chalendar, I., Partington, J.R.: Analyticity and compactness of semigroups of composition operators. J. Math. Anal. Appl. 437, 545–560 (2016) 7. Bayart,F.:Hardy spaces ofDirichletseries andtheircomposition operators. Monatsh.Math. 136, 203–236 (2002) 8. Bayart, F., Castillo-Medina, J., García, D., Maestre, M., Sevilla-Peris, P.: Composition operators on spaces of double Dirichlet series. Rev. Mat. Complut. 34, 215–237 (2021) 9. Berkson, E., Porta, H.: Semigroups of analytic functions and composition operators. Mich. Math. J. 25, 101–115 (1978) 10. Betsakos, D.: On the eigenvalues of the infinitesimal generator of a semigroup of composition operators. J. Funct. Anal. 273, 2249–2274 (2017) 11. Blasco, O., Contreras, M.D., Díaz-Madrigal, S., Martínez, J., Papadimitrakis, M., Siskakis, A.G.: Semigroups of composition operators and integral operators in spaces of analytic functions. Ann. Acad. Sci. Fenn. Math. 38, 67–89 (2013) 12. Bohr, H.:Über diegleichmässige KonvergenzDirichletscher Reihen.J.ReineAngew.Math. 143,203–211 (1913) 13. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Continuous Semigroups of Holomorphic Self-Maps of the Unit Disc. Springer, Berlin (2020) 14. Brevig, O., Kouroupis, A.: An extension of Bohr’s theorem, to appear in Proc. Amer. Math. Soc. arXiv:2302.14519 15. Contreras, M.D., Díaz-Madrigal, S.: Compact-type operators defined on H∞. Contemp. Math. 232, 111–118 (1999) 16. Contreras, M.D., Díaz-Madrigal, S.: Fractional iteration in the disk algebra: prime ends and composition operators. Rev. Mat. Iberoam. 21, 911–928 (2005) 17. Contreras, M.D., Gómez-Cabello, C., Rodríguez-Piazza, L.: On the uniform convergence of continuous semigroups, preprint. arXiv:2304.12759 18. Contreras, M.D., Gómez-Cabello, C., Rodríguez-Piazza, L.: Semigroups of composition operators on Hardy spaces of Dirichlet series. J. Func. Anal. 285, 1–36 (2023) 19. Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Dirichlet Series and Holomorphic Functions in High Dimensions. Cambridge University Press, Cambridge (2019) 20. Engel, K.J., Nagel, R.: A Short Course on Operator Semigroups. Springer, New York (2006) 21. Gallardo-Gutiérrez, E.A., Yakubovich, D.V.: On generators of C0-semigroups of composition operators. Israel J. Math. 229, 487–500 (2019) 22. Gamelin, T.W.: Complex Analysis. Undergraduate Texts in Mathematics. Springer, Berlin (2001) 123