The Cesàro space of Dirichlet series.

Programa de Doctorado “Matem´aticas” PHD DISSERTATION The Ces`aro space of Dirichlet series Author Jos´e Jorge Bueno Contreras Supervisors Dr. Guillermo P. Curbera Costello Dr. Olvido Delgado Garrido Acknowledgements First and foremost, I would like to thank my advisor, Guillermo Curbera. I am greatly indebted to him for his patience, motivation and great support over these past four years. Not forgotten, my appreciation to my co-advisor, Olvido Garrido, for her contributions to this research and her terrific work of review and correction. This thesis would have never seen the light without their assistance and dedicated involvement. I am also grateful to Professor Pascal Lef`evre, for all the aid he provided during my stay at Universit´e d’Artois, from February to April 2017, and his insightful remarks on the subject of the thesis. A special mention goes to professors Helga Fetter and Berta Gamboa, from CIMAT in Guanajuato, for initiating me in the arts of mathematical analysis. To my Mexican friends Ulises (who is indirectly responsible for me ending up in Spain), Sauri and Ricardo, for hosting me during some of my travels and coming to see me. And to Malors, for keeping in touch after all this time and making me look for books and notebooks wherever I went. I also owe my sincere thanks to the people I have met in Seville, without whom I wouldn’t have survived for so long. Rodiak was the one who delivered me from the initial solitude after my arrival, and Diego and Jaqueline also made quite an effort in keeping my social life actually alive. We visited several places together and I am very glad for meeting them. Jeroen was my other travel companion, with whom I went to some of the most distant Andalusian provinces. Lunchtime would have been terribly lame without Cristian and Soledad (and everyone before in this paragraph!). We still have some pending escape rooms. I am also very grateful to Juampe, for somehow making me talk quite a lot... whoever knows me should understand the magnitude of that achievement. Last but not the least, I would like to thank my family: my father Jorge, my mother Ofelia and my sisters Ana and Rub´ı. Thanks for your love and unconditional support. I wish you could be here for the defence. Contents Introducction 7 Notation 13 1 Preliminaries 15 1.1 The study of Banach spaces of Dirichlet series . . . . . . . . . 15 1.2 Dirichletseries .......................... 21 1.3 Hardy spaces of analytic functions on the unit disc . . . . . . 28 1.4 Banach lattices of sequences . . . . . . . . . . . . . . . . . . . 29 2 The Ces`aro space of Dirichlet series 33 2.1 The Ces`aro sequence space cesp................. 34 2.2 The Ces`aro space of Dirichlet series H(cesp) .......... 40 2.3 The Banach spaces of Dirichlet series Ar,A2,r and H∞(Cr) . . 52 2.4 Other Banach spaces of Dirichlet series . . . . . . . . . . . . . 59 3 The multiplier algebra of H(cesp)65 3.1 General facts on multiplier algebras . . . . . . . . . . . . . . . 66 3.2 First examples of multiplier algebras . . . . . . . . . . . . . . 71 3.3 The multiplier algebra of H(cesp) ................ 75 3.4 Further facts on multipliers on H(cesp)............. 93 4 The Ces`aro operator acting on Dirichlet series 99 Bibliography 103 Introduction This memoir is devoted to the interplay between two different mathematical objects: spaces of Dirichlet series and the sequence space cesp. Several spaces of Dirichlet series have been studied in recent years. Hedenmalm, Lindqvist and Seip defined in [34] the Hilbert space of Dirichlet series, H, consisting of all Dirichlet series f(s) := ∞ ∑ n=1 ann−s, s ∈C, with square summable coefficients. They used it for solving a problem discussed by Beurling on complete sequences in the space L2(0,1). Due to the Cauchy-Schwarz inequality, each f∈ H defines an analytic function on the vertical half-plane C1/2:= {s∈C:ℜ(s)>1/2}. The space Hbecomes a Banach space of analytic functions on C1/2when endowed with the norm ∥f∥H:= ∥(an)∞ n=1∥ℓ2, f ∈ H. The Hardy spaces of Dirichlet series Hp, for 1 ≤p < ∞, were introduced by Bayart in [10]. They are analogous, in some sense, to the classical Hardy spaces Hp(D) on the unit disc Dof the complex plane. Namely, Hpis the completion of the space of Dirichlet polynomials P(s) := ∑N n=1 ann−sfor the norm ∥P∥Hp:= (lim T→∞ 1 2T∫T −T|P(it)|pdt)1/p . The space Hcorresponds to Hpfor p= 2. In [44], McCarthy considered the weighted Hilbert spaces of Dirichlet series Hα:= {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=2 |an|2(log n)−α<∞}, for α > 0, endowed with the norm ∥f∥Hα:= ∥(an(log n)−α/2)∞ n=2∥ℓ2, f ∈ Hα. 7 8Introduction More recently, Bailleul and Lef`evre have studied certain classes of Bergmantype spaces of Dirichlet series, Ap µand Bp, for 1 ≤p < ∞, [8]. Also, another type of weighted Hilbert spaces of Dirichlet series Dα, for α > 0, has been considered by Bailleul and Brevig in [7]. It is noticeable that the spaces H, Hp,Hα,Ap µ,Bp,Dαare all Banach spaces of analytic functions on the vertical half-plane C1/2. A particularly difficult and deep feature of Dirichlet series is their product. The pointwise product f(s)·g(s) of two Dirichlet series f(s) = ∑∞ n=1 ann−sand g(s) = ∑∞ n=1 bnn−sis the Dirichlet series h(s) = ∑∞ n=1 cnn−swhose coefficients c= (cn)∞ n=1 are given by the Dirichlet convolution c:= a·bof the sequences a= (an)∞ n=1 and b= (bn)∞ n=1, that is, cn= (a·b)n:= ∑ k|n akbn k, n ≥1, where k|ndenotes that kis a divisor of n. Given a space Eof Dirichlet series converging on a common vertical halfplane Cθ, a multiplier on Eis an analytic function fon Cθwith the property that fg ∈ E for every g∈ E. The multiplier algebra of Eis the space of all multipliers on E; we denote it by M(E). Note that M(E)⊆ E whenever 1∈ E. Neither of the spaces H,Hp,Hα,Ap µ,Bp,Dαis closed under multiplication, hence a relevant question is to identify their multiplier algebras. Hedenmalm, Lindqvist and Seip identified the multiplier algebra Mof the Hilbert space of Dirichlet series Hproving that M=H∞, where H∞is the algebra of bounded analytic functions on C0which can be represented as a Dirichlet series in some vertical half-plane, [34, Theorem 3.1]. This identification was a key step in solving the Beurling’s question on complete sequences. It turns out that for all the spaces Hp,Hα,Ap µ,Bp,Dα, the multiplier algebra is also the algebra H∞; [10, Theorem 7], [44, Theorem 1.11] [4, Theorem 10.1 and Theorem 11.21], [7, Theorem 3]. We now turn to the sequence space cesp. For 1 < p < ∞, Hardy’s inequality, ∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p <(p p−1)p∞ ∑ n=1 |an|p, [32, Theorem 326], established the boundedness on ℓpof the Ces`aro averaging operator C, defined by a= (an)∞ n=1 7−→ C(a) := (1 n n ∑ k=1 ak)∞ n=1 . Introduction 9 The Ces`aro sequence space cespis defined, in a natural way, as cesp:= {(an)∞ n=1 ∈CN:∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p <∞}. It is a linear space which becomes a Banach space under the norm ∥(an)∞ n=1∥cesp:= (∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p)1/p ,(an)∞ n=1 ∈cesp. Hardy’s inequality shows that ℓpis continuously contained in cesp. The contention is proper since cespcontains sequences with arbitrarily large terms; this is an important feature of cespthat will be detailed in Section 2.1 below. The space cesphas been thoroughly studied by G. Bennett, [14]. In this work we study the Ces`aro space of Dirichlet series H(cesp), for 1< p < ∞, consisting of all Dirichlet series whose sequence of coefficients belongs to the Ces`aro sequence space cesp, that is, H(cesp) := {f(s) = ∞ ∑ n=1 ann−s: (an)∞ n=1 ∈cesp}. It is a linear space, which becomes a Banach space of Dirichlet series when endowed with the norm ∥f∥H(cesp):= ∥(an)∞ n=1∥cesp=(∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p)1/p , f ∈ H(cesp). The space H(cesp) contains Dirichlet series whose coefficients can be arbitrarily large (as cespso does). This fact establishes a technical difference for studying H(cesp), when compared with previously considered spaces of Dirichlet series. We place special emphasis on identifying the multiplier algebra M(H(cesp)) of H(cesp). After this Introduction and some Notation, the memoir is organized into four chapters. Chapter 1contains preliminary materials and is divided into four sections. The first section presents a brief account of the main works devoted to studying Banach spaces of Dirichlet series, since Hedenmalm, Lindqvist and Seip defined the Hilbert space of Dirichlet series Hin their seminal 1997 paper, [34]. The second section collects basic results regarding Dirichlet series and their convergence properties. The third section is devoted to briefly summarize basic facts about Hardy spaces of analytic functions on the unit disc. 16 Chapter 1. Preliminaries φits dilated system forms a weaker type of basis? They focused on finding conditions under which the system (φn)∞ n=1 is a Riesz basis, that is, it is complete in L2(0,1) and there exist positive constants Aand Bsuch that A(∞ ∑ n=1 |cn|2)1/2 ≤    ∞ ∑ n=1 cnφn   L2(0,1) ≤B(∞ ∑ n=1 |cn|2)1/2 for every (cn)∞ n=1 ∈ℓ2. The original question was proposed by Beurling and involves only completeness, [15]. He pointed out a possible way to study this problem consisting in the use of appropriated Dirichlet series. Considering the orthonormal basis (√2 sin(nπx))∞ n=1 of L2(0,1), the function φcan be written as φ(x) = ∞ ∑ n=1 an√2 sin(nπx),∞ ∑ n=1 |an|2<∞, and then it can be associated to the Dirichlet series Sφ(s) := ∞ ∑ n=1 ann−s. Through this approach, Hedenmalm, Lindqvist and Seip found a simple characterization of Riesz basis. They also considered the problem of only completeness and gave equivalent cyclicity conditions, but they are not so fine. Theorem 1.1 (Hedenmalm, Lindqvist, Seip).The dilated system (φn)∞ n=1 is a Riesz basis if and only if both Sφ and 1/Sφ are bounded analytic functions on the vertical half-plane C0. The spaces of Dirichlet series initially introduced in [34] as part of the solution to the Riesz basis problem were Hand H∞. The space His the space of all Dirichlet series with square-summable coefficients, that is, H:= {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 |an|2<∞}. Considering the inner product of two series in Has the inner product of their coefficient sequences as elements of ℓ2, we have that His a Hilbert space with norm ∥f∥H:= (∞ ∑ n=1 |an|2)1/2 , f ∈ H. From the Cauchy-Schwarz inequality it follows that ∞ ∑ n=1  an ns≤(∞ ∑ n=1 |an|2)1/2(∞ ∑ n=1 1 n2ℜ(s))1/2 Chapter 1. Preliminaries 17 and so every f∈ H converges absolutely in the vertical half-plane C1/2. Then, His a space of analytic functions on C1/2, which is in fact an optimal domain (see Proposition 2.23.(a)). The space H∞consists of all Dirichlet series convergent at some point, which define, possibly through analytic continuation, a bounded analytic function on the vertical half-plane C0. This space turns out to be a Banach space under the norm ∥f∥H∞:= sup s∈C0|f(s)|, f ∈ H∞ (see [45, Theorem 6.2.1]). Although the definition of H∞does not require convergence in all C0of the Dirichlet series associated to a function f∈ H∞, this actually happens due to the boundedness and analyticity of fand the application of a Bohr’s theorem (see Theorem 1.15). Theorem 1.2. For every f∈ H∞, the associated Dirichlet series of fconverges uniformly in each vertical half-plane Cϵwith ϵ > 0. As a consequence of the previous theorem and the Carlson’s identity [22] a first connection between the spaces Hand H∞is found. Theorem 1.3. For every f(s) = ∑∞ n=1 ann−s∈ H∞and ϵ > 0, it follows that ∞ ∑ n=1 |an|2n−2ϵ= lim T→∞ 1 2T∫T −T|f(ϵ+it)|2dt ≤ ∥f∥2 H∞ In particular, H∞⊆ H continuously with ∥f∥H≤ ∥f∥H∞for f∈ H∞. The relationship between Hand H∞is actually deeper than the inclusion stated above, involving the concept of multiplier. A multiplier on His an analytic function fon C1/2such that fg ∈ H for every g∈ H. One of the main results in [34] states that the space Mof all multipliers on Hcoincides with H∞; in fact, after this result, Theorem 1.1 is proved for Sφ and 1/Sφ belonging to M. Theorem 1.4 (Hedenmalm, Lindqvist, Seip).The equality M=H∞holds with ∥f∥M:= sup g∈H ∥g∥H≤1 ∥fg∥H=∥f∥H∞, f ∈ M. Note that each f∈ M defines a bounded multiplication operator on H and ∥f∥Mis just its operator norm. Even more, Mis a closed subspace of the space of all bounded linear operators on H(see Proposition 3.1). 18 Chapter 1. Preliminaries Theorem 1.4 is the analogous of Schur’s result which identifies the multipliers on the Hardy space H2(D) (see Theorem 1.24). Indeed, althoug there are other proofs (see [45, Section 6.4.3]), the original one by Hedenmalm, Lindqvist and Seip is based on the identification of the spaces Hand H∞ with suitable Hardy spaces of analytic functions. The identification is made through a technique known as the Bohr’s lift which was introduced by Bohr in [18]. Consider a Dirichlet series f(s) = ∑∞ n=1 ann−s. The fundamental theorem of arithmetic guarantees that each n∈Ncan be written, in an unique way, as n=pα1 1pα2 2. . . pαk k, where (pr)∞ r=1 is the sequence of the prime numbers written in increasing order and αr≥0. Then f(s) = ∞ ∑ n=1 an(pα1 1pα2 2. . . pαk k)−s=∞ ∑ n=1 an(p−s 1)α1(p−s 2)α2. . . (p−s k)αk. By thinking of the prime powers p−s ias independent variables zi, the series f can be considered as a power series in infinite variables: D(f) := ∞ ∑ n=1 anzα1 1zα2 2. . . zαk k.(1.1) Note that zi∈Dwhenever s∈C0. One of the key points in the proof of Theorem 1.4, and probably the hardest to prove, is that the Bohr’s lift establishes an isometric isomorphism between H,H∞and the Hardy spaces on the infinite-dimensional polydisk H2(D∞), H∞(D∞), respectively. After this, the problem is reduced to proving that M(H2(D∞)) = H∞(D∞). In 2002, as a generalization of H, Bayart introduced the Hardy spaces of Dirichlet series Hp, for 1 ≤p < ∞, [10] and [11]. Since the norm of the Hardy space Hp(D) is obtained by integration over circles approaching the unit circle (see Section 1.3), to obtain the analogous for Dirichlet series whose domains are vertical half-planes, it seems natural to integrate on vertical lines, just as it is done in Theorem 1.3. For Pbelonging to the space Pof Dirichlet polynomials ∑N n=1 ann−s, set ∥P∥Hp:= (lim T→∞ 1 2T∫T −T|P(it)|pdt)1/p . The space Hpis defined as the completion of Punder the norm ∥ · ∥Hp. Using the Bohr’s lift, Bayart proved that Hpis a Banach space isometrically Chapter 1. Preliminaries 19 isomorphic to the Hardy space on the infinite-dimensional polycircle Hp(T∞) as defined in [23] by Cole and Gamelin. After this, the following properties are obtained for Hp: each function in Hpcan be represented by a Dirichlet series, Hpis a space of analytic functions on C1/2(which is a maximal domain) and the point evaluation δs0at a point s0∈C1/2is bounded on Hpwith norm ∥δs0∥=ζ(2ℜ(s0))1/p. Moreover, the Hardy space H2is just the Hilbert space Hand Theorem 1.4 is extended to Hp. Theorem 1.5 (Bayart).The space of multipliers on Hpcoincides isometrically with H∞. More recently, in 2014, Aleman, Olsen and Saksman proved other important result: for 1 < p < ∞, the sequence of monomials (n−s)∞ n=1 is a Schauder basis of Hp, [1]. This means that the partial sums of a Dirichlet series f∈ Hp converges to fin Hp. In 2004, McCarthy studied the weighted Hilbert spaces Hw:= {f(s) = ∞ ∑ n=n0 ann−s:∞ ∑ n=n0|an|2wn<∞}, where n0∈Nand w= (wn)∞ n=n0is a sequence of positive numbers, [44]. These spaces are Banach spaces with norm ∥f∥Hw:= (∞ ∑ n=n0|an|2wn)1/2 . Of course, His a particular case obtained for w= (1)∞ n=1. The sequence of weights wmust be defined in a particular way to ensure a good behavior of Hwas a space of Dirichlet series: let µbe a positive Radon measure on [0,∞) such that 0 is in the support of µand ∫∞ 0 n−2σ 0dµ(σ)<∞. Then the weights in ware taken as wn:= ∫∞ 0 n−2σdµ(σ), n ≥n0. For this w, the space Hwis a space of analytic functions on C1/2, which is an optimal domain, and again the same result is obtain for its multiplier algebra, although in this time McCarthy shows a different proof without Bohr’s lift. Theorem 1.6 (McCarthy).The space of multipliers on Hwcoincides isometrically with H∞. 20 Chapter 1. Preliminaries Of particular relevance is the space Hwobtained for the measure µgiven by dµ(σ)=2αΓ(α)−1σα−1dσ with α > 0. In this case w= ((log n)−α)∞ n=2 produces the space Hα:= {f(s) = ∞ ∑ n=2 ann−s:∞ ∑ n=2 |an|2(log n)−α<∞}. In 2015, Bailleul and Lef`evre introduced two Bergman-type spaces of Dirichlet series, Ap µand Bp, for 1 ≤p < ∞, [8]. For the construcction of Ap µ, they consider a probability measure µon (0,∞) such that 0 is in the support of µ, and for Pbelonging to the space Pof Dirichlet polynomials, define ∥P∥Ap µ:= (∫∞ 0∥Pσ∥p Hpdµ(σ))1/p , where Pσ∈ P is given by Pσ(s) = P(s+σ) for s∈C. Then, they take the space Ap µas the completion of Punder the norm ∥·∥Ap µ. Note that A2 µ=Hw for w= (∫∞ 0n−2σdµ(σ))∞ n=1. For the space Bp, Bailleul and Lef`evre consider the measure A=λ⊗λ⊗··· on the infinite-dimensional polydisk D∞, where λis the normalized Lebesgue measure on D, and for P∈ P, define ∥P∥Bp:= (∫D∞|D(P)|pdA)1/p , where D(P) is the analytic polynomial associated to Pvia (1.1). Then, Bp is the completion of Pwith respect to the norm ∥·∥Bp. Both Ap µand Bpturn out to be spaces of analytic functions on C1/2which actually are Dirichlet series. Bailleul found in his thesis [4] the same result for the multiplier algebras of these spaces. Theorem 1.7 (Bailleul).The space of multipliers on Ap µand the space of multipliers on Bpcoincide isometrically with H∞. A year later, Bailleul and Brevig considered the weighted Hilbert space Dα:= {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 |an|2 [d(n)]α<∞}. where α∈Rand d(n) denotes the number of divisors of the integer n, [7]. Endowed with the norm ∥f∥Dα=(∞ ∑ n=1 |an|2 [d(n)]α)1/2 , Dαis a Banach space of analytic functions on C1/2. Chapter 1. Preliminaries 21 Theorem 1.8 (Bailleul, Brevig).For α > 0, the space of multipliers on Dα coincide isometrically with H∞. Athough we focus our attention on the multiplier algebra, there are other important issues studied for spaces of Dirichlet series, as for instance the composition operator. In 1999, Gordon and Hedenmalm proposed and solved the following problem, [29]: for which analytic functions Φ : C1/2→C1/2does the composition operator CΦ:f→ CΦ(f) = f◦Φ define a bounded operator from Hinto itself? They gave the following characterization for such a Φ. Theorem 1.9 (Gordon-Hedenmalm).An analytic function Φ : C1/2→C1/2 defines a bounded composition operator CΦ:H → H if and only if (a) Φ(s) = c0s+φ(s)with c0∈N∪{0}and φbeing Dirichlet series converging at some point, and (b) Φhas an analytic extension to C0, also denoted by Φ, such that (i) Φ(C0)⊂C0if c0= 0, and (ii) Φ(C0)⊂C1/2if c0= 0. Similar results have been obtained for composition operators on Hp[10], Ap µ[5], Bp,Dαand Hα[7]. Several properties as compactness or invertibility have also been studied in these and other papers, see [11], [12], [28], [13], [6], [4]. 1.2 Dirichlet series We include in this section a summary of basic results regarding Dirichlet series and their convergence properties. There are excellent works for further and detailed study of the subject: •H. Bohr’s doctoral dissertation, Bidrag til de Dirichletske Rækkers Theori (Contributions to the Theory of Dirichlet Series), 1910, [17]. •The classical book by G. H. Hardy and M. Riesz, The General Theory of Dirichlet’s Series, Cambridge University Press, Cambridge, 1915, [33], •Chapter IX of the book by E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1932, [49]. 22 Chapter 1. Preliminaries •Chapter 11 of the book by T. M. Apostol Introduction to Analytic Number Theory, Springer-Verlag, New York-Heidelberg, 1976, [2]. •The recent book by H. Queff´elec and M. Queff´elec Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013, [45]. A Dirichlet series is a series of the form f(s) = ∞ ∑ n=1 ann−s,(1.2) where a= (an)∞ n=1 is a given complex sequence, and sis a complex variable. In this work we will write s=σ+it, following Riemann’s notation, where σ=ℜ(s) and t=ℑ(s) are real numbers. These series are not of such importance in general analysis as power series since they only represent a very special class of analytic functions. They are, however, of great relevance in the applications of analysis to number theory. As their name implies, they were first introduced by P. G. Lejeune Dirichlet (his given name was Peter Gustav, and Lejeune Dirichlet was his surname) as part of the proof of his classical theorem on arithmetic progressions [42]. Dirichlet considered only real values for the variable s. Dirichlet series were later used by Hadamard, [30], and De la Vall´ee-Poussin, [26], in their proofs for the Prime Number Theorem. According to Hardy, [33, p.1]: The first theorems involving complex values of sare due to Jensen, who determined the nature of the region of convergence of the general series; and the first attempt to construct a systematic theory of the function f(s) was made by Cahen in a memoir which, although much of the analysis which it contains is open to serious criticism, has served–and possibly just for that reason–as the starting point of most of the later researches in the subject. Apart from the work by Jensen [38,39] and Cahen [21], other ideas and results regarding the convergence theory of Dirichlet series can be found in the earlier works of Dedekind [43], Kronecker [40] and Stieltjes [48]. The theory of Dirichlet series is more complicated than the theory of power series. Indeed, one of the simplest power series is ∑∞ n=0 zn(all coefficients equal to one) which converges absolutely on the unit disc and can be analytically Chapter 1. Preliminaries 23 extended to C\{1}, while the corresponding Dirichlet series is ∑∞ n=1 n−swhich defines the Riemann zeta function! Great differences appear in the matter of convergence regions. Power series are convergent in open discs (unless their radius of convergence is equal to 0), where they are absolutely convergent. The function defined by the series is bounded and the partial sums converge uniformly to the function on any smaller disc with the same center. The natural regions of convergence for Dirichlet series are vertical half-planes instead of discs and, unlike the case of power series, the behaviour previously described may occur in different vertical half-planes. Theorem 1.10 (Jensen’s lemma).If a Dirichlet series is convergent for s= s0, then it is uniformly convergent on each cone SC={s: 0 ≤|s−s0| ℜ(s−s0)≤C}, C ≥1. From now on, for θ∈R, we will denote by Cθthe vertical half-plane Cθ:= {s∈C:ℜ(s)> θ}. An immediate consequence of the previous theorem is that any Dirichlet series convergent at some point s0=σ0+it0∈Cis actually convergent in all the vertical half-plane Cσ0, where it defines an analytic function. The following theorem directly involves convergence on vertical half-planes, although it’s only pointwise convergence. Theorem 1.11 (Bohr).Given a Dirichlet series ∑∞ n=1 ann−s, denote its partial sums by SN(s) = ∑N n=1 ann−s. If the sequence {SN(s0)}∞ N=1 is bounded for some s0∈C, then the series is convergent in the vertical half-plane ℜ(s)>ℜ(s0). Taking the previous results into account, it makes sense to define the abscissa of convergence of a Dirichlet series fas σc(f) = inf {θ: the series (1.2) converges in Cθ}. The existence of this abscissa was first noticed by Jensen in [38]. This number may be +∞if the series is nowhere convergent, or −∞ if the series converges in the whole complex plane. Hence, (a) The series (1.2) is convergent and defines an analytic function in Cσc(f). 24 Chapter 1. Preliminaries (b) The series (1.2) is divergent in the vertical half-plane ℜ(s)< σc(f). The line ℜ(s) = σc(f) is known as the boundary of convergence of f. Not much can be said about the convergence of a series on that line since it may converge or diverge in the whole line, or be convergent in some points and divergent in the rest. Note that the existence of some s0∈Csuch that the partial sums ∑N n=1 ann−s0are bounded but not convergent, can only occur for points on the boundary of convergence. Theorem 1.12 (Cahen).Let f(s) = ∑∞ n=1 ann−sbe a Dirichlet series. If there exist s0∈Cand K > 0such that |ann−s0| ≤ Kfor all n∈N, then the series converges absolutely in the vertical half-plane Cσ0+1, where σ0=ℜ(s0). The previous result shows us that a somewhere convergent Dirichlet series is absolutely convergent in all vertical half-planes Cθfor θbig enough. This inspires the definition of the abscissa of absolute convergence: σa(f) = inf {θ: the series (1.2) converges absolutely in Cθ}. Clearly σc(f)≤σa(f) and fconverges absolutely in the vertical half-plane ℜ(s)> σa(f). Unlike the case of conditional convergence, if a Dirichlet series converges at any point of its boundary of absolute convergence then it converges on the whole line. As we mentioned at the beginning of this chapter, there may exist a region where the series is convergent but not absolutely convergent. An example of this is Riemann’s alternating zeta function, also known as Dirichlet’s eta function: η(s) = ∞ ∑ n=1 (−1)nn−s,(1.3) which is convergent in the vertical half-plane C0but only converges absolutely in C1. The natural question about how big is this strip of conditional convergence is answered in the previous theorem: σa(f)−σc(f)≤1. Even more, the constant 1 cannot be improved, as it is attained in the series (1.3). Let us denote by Dthe set of all Dirichlet series that are convergent at some point; this can be equivalently defined as the set of all Dirichlet series such that the sequence of its coefficients has, at most, a polynomial growth rate. Indeed, if ∑∞ n=1 ann−s0is convergent then it converges absolutely for some integer s=N > ℜ(s0) + 1, so |ann−N| ≤ 1 for nbig enough, which immediately implies the polynomial growth of an. The sufficiency of that condition is obtained in a similar way. Just like the case of power series, if an analytic function can be written as a Dirichlet series such representation is unique, as stated in the following theorem. Chapter 1. Preliminaries 25 Theorem 1.13 (Dirichlet-Dedekind).Let f(s) = ∑∞ n=1 ann−sand g(s) = ∑∞ n=1 bnn−sbe in D. If there exists a sequence of complex numbers (sn)∞ n=1 such that ℜ(sn)→ ∞ as n→ ∞, and f(sn) = g(sn)for all n∈N, then the series ∑∞ n=1 ann−sand ∑∞ n=1 bnn−sare identical, i.e., we have an=bnfor all n∈N. In the case of power series the Cauchy-Hadamard theorem gives us an expression for the radius of convergence of a series in terms of its coefficients. So, it is natural to look for an analogous expression for the convergence abscissa of a Dirichlet series. The corresponding formula takes slightly different forms according to whether ∑anis convergent or not. Theorem 1.14 (Cahen-Titchmarsh).Let f(s) = ∑∞ n=1 ann−s. If ∑∞ n=1 an diverges, then σc(f) = lim sup N→∞ log ∑N n=1 an log N. On the other hand, if ∑∞ n=1 anis a convergent series, we have σc(f) = lim sup N→∞ log ∑∞ n=N+1 an log N. The formula for the first case is due to Cahen, [21], and regarding the second one, which is much less known, the only reference we have found is Titchmarsh’s book, [49]. Note that, if f(s) = ∑∞ n=1 ann−sand g(s) = ∑∞ n=1 |an|n−s, then σa(f) = σc(g), so σa(f) = lim sup N→∞ log ∑N n=1 |an| log N if ∑∞ n=1 |an|=∞, and σa(f) = lim sup N→∞ log ∑∞ n=N+1 |an| log N whenever ∑∞ n=1 |an|<∞. Observe that if a Dirichlet series converges absolutely for some s0∈C, the Weierstrass M-test implies its uniform convergence in the vertical half-plane Cσ0. Therefore, it makes sense to define the abscissa of uniform convergence as σu(f) = inf {θ: the series (1.2) converges uniformly in Cθ}, and clearly σu(f)≤σa(f). Chapter 2 The Ces`aro space of Dirichlet series This chapter is devoted to the study of the Ces`aro space of Dirichlet series H(cesp), for 1 <p<∞. It is organized into three sections. In the first section we review the origin, definition and properties of the sequences space cesp. This space contains sequences with arbitrarily large terms (Example 2.1), although there is control on the growth of the partial sums (Proposition 2.2). The canonical vectors constitute a well behaved Schauder basis (Proposition 2.9). We present the isomorphic identification of the Banach dual space given by Jagers (Theorem 2.5) and the isomorphic identification by given G. Bennett (Theorem 2.6). In the second section we define the Ces`aro space of Dirichlet series H(cesp) and study its basic properties. We show the existence of a well behaved Schauder basis formed by the monomials n−s, for n≥1 (Proposition 2.10), and we show the approximation of functions in H(cesp) by partial series obtained by restricting the number of prime numbers in the summation indices (Proposition 2.11). We show that H(cesp) is a space of analytic functions on the vertical half-plane C1/q, for 1/p + 1/q = 1, by calculating the convergence and absolute convergence abscissae (Theorem 2.12), and we study the boundedness of the point evaluation functionals on H(cesp), giving sharp estimates for their norm and the precise order of growth when the abscissa approaches the critical value 1/q (Theorem 2.13). For the case p= 2 some additional information is available on the norm of the point evaluations and on equivalent expressions for the norm in H(ces2). In the third section we discuss other Banach spaces of Dirichlet series that will appear in the memoir: Ar,A2,r and H(Cr). We calculate their convergence abscissae and prove the boundedness of point evaluations. We prove that Aris the solid core of H(Cr) (Proposition 2.21) and that Dirichlet series in H(Cr) with real coefficients where the sign behaves multiplicatively in fact belong to Ar 33 34 Chapter 2. The Ces`aro space of Dirichlet series (Theorem 2.22). We end exhibiting the spaces of Dirichlet series H(ℓp) and H([C, ℓp]). 2.1 The Ces`aro sequence space cesp The Ces`aro sequence space cespwas first introduced in 1968 when the Dutch Mathematical Society proposed the identification of its dual space as a prize problem, [50]. Jagers solved the problem in a more general context, and gave an explicit isometric identification of the dual space, [37]. In 1996, G. Bennett made a thorough study of the spaces cespin the memoire entitled “Factorizing the Classical Inequalities”, [14]. Astashkin and Maligranda have also studied these spaces, see, for example, [3], and the references therein. For 1 < p < ∞, the Ces`aro sequence space cespis defined as cesp:= {x= (xn)∞ n=1 ∈CN:∞ ∑ n=1 (1 n n ∑ k=1 |xk|)p <∞}. It is a linear space which becomes a Banach space under the norm ∥x∥cesp:= (∞ ∑ n=1 (1 n n ∑ k=1 |xk|)p)1/p , x ∈cesp. Note that 1 < p1≤p2<∞implies cesp1⊆cesp2. The space cesparises in a natural way from Hardy’s inequality, ∞ ∑ n=1 (1 n n ∑ k=1 |xk|)p <(p p−1)p∞ ∑ n=1 |xn|p, [32, Theorem 326], which establishes the boundedness on ℓpof the Ces`aro averaging operator Cdefined by x= (xn)∞ n=1 7−→ C(x) := (1 n n ∑ k=1 xk)∞ n=1 . Hardy’s inequality shows that ℓpis continuously contained in cesp. The averaging process appearing in the definition of cesp“softens” the growth of sequences, allowing some unbounded sequences to belong to cesp. In fact, cesp contains sequences with arbitrarily large terms. Chapter 2. The Ces`aro space of Dirichlet series 35 Example 2.1. Given any complex sequence a= (am)∞ m=1 and an increasing sequence of natural numbers 1 < N1< N2< . . ., define xk:= {amif k=Nm 0 otherwise . It is possible to choose a sequence (Nk)∞ k=1 so that x= (xk)∞ k=1 belongs to cesp. Indeed, from the definition of xit follows that Nm+1−1 ∑ k=Nm 1 kp(k ∑ j=1 |xj|)p = Nm+1−1 ∑ k=Nm 1 kp(m ∑ i=1 |ai|)p ≤(m ∑ i=1 |ai|)p∞ ∑ k=Nm 1 kp ≤(m ∑ i=1 |ai|)p1 p−1·1 (Nm−1)p−1. Taking (Nk)∞ k=1 increasing fast enough, in a way that the last term of the above inequality is appropriately small, we have that ∞ ∑ m=1 1 kp(k ∑ j=1 |xj|)p =∞ ∑ m=1 Nm+1−1 ∑ k=Nm 1 kp(k ∑ j=1 |xj|)p <∞. The previous example is specially interesting when the sequence ais not bounded, showing us that the space cespis much larger than ℓp. Nevertheless there is still some control on the growth of the partial sums of elements in cesp, as the next result shows, [25, Proposition 2.3(iii)]. We include a proof for completeness. Recall that we denote by qthe conjugate exponent of p. Proposition 2.2 (Curbera, Ricker).If (xn)∞ n=1 ∈cespthen lim n→∞ 1 n1/q n ∑ k=1 |xk|= 0. Proof. We have that ∞ ∑ k=n(1 k k ∑ j=1 |xj|)p ≥∞ ∑ k=n(1 k n ∑ j=1 |xj|)p =(n ∑ j=1 |xj|)p∑ k≥n 1 kp ≥(n ∑ j=1 |xj|)p1 (p−1)np−1. 36 Chapter 2. The Ces`aro space of Dirichlet series Since the first term of the above inequality tends to zero when n→ ∞ as (xn)∞ n=1 ∈cesp, it follows that lim n→∞ 1 np−1(n ∑ j=1 |xj|)p = 0. Raising to the power 1/p, as (p−1)/p = 1/q, we obtain the result. Jagers studied in [37] a family of spaces more general than cesp, namely the spaces bp, for 1 ≤p < ∞. Given a sequence β= (βn)∞ n=1 of positive numbers, the space bpassociated to βconsists of all sequences x= (xn)∞ n=1 satisfying ∥x∥bp:= (∞ ∑ n=1 (βn n ∑ k=1 |xk|)p)1/p <∞. The particular case of β= (n−1)∞ n=1 gives the Ces`aro sequence space cesp. It is immediate that bp={0}if and only if ∑∞ n=1 βp n=∞. Hence, it is natural to assume that ∑∞ n=1 βp n<∞. We denote Bm:= ∞ ∑ k=m βp k, m ≥1. The space bpis a Banach space endowed with the norm ∥·∥bp. Proposition 2.3. For each m≥1, the m-th coordinate functional πmon bp, given by πm(x) := xmfor x= (xn)∞ n=1 ∈bp, is bounded and ∥πm∥=B−1 p m. Proof. For every x= (xn)∞ n=1 ∈bpwe have that ∥x∥p bp≥∞ ∑ n=m(βn n ∑ k=1 |xk|)p ≥ |xm|pBm and so |πm(x)|≤∥x∥bpB−1 p m, which gives ∥πm∥ ≤ B−1 p m. On the other hand, note that the canonical vectors (en)∞ n=1 are included in bpwith ∥en∥bp=B 1 p n. Then, taking x=em/∥em∥bpwe obtain that B−1 p m=|πm(x)| ≤ ∥πm∥. General properties of bpare collected in the next proposition, see [37]. Proposition 2.4 (Jagers).The following statements hold: Chapter 2. The Ces`aro space of Dirichlet series 37 (a) For every x= (xn)∞ n=1 ∈bpit follows that ∑N n=1 xnenconverges to x in the norm of bp. In particular, as the m-th coordinate functionals are continuous, the canonical vectors (en)∞ n=1 form a Schauder basis. (b) The dual Banach space b∗ pof bpcan be identified with the sequence space {(un)∞ n=1 ∈CN:∞ ∑ n=1 |xnun|<∞for all (xn)∞ n=1 ∈bp},(2.1) via u∗∈b∗ p7→ (u∗(en))∞ n=1. (c) For 1< p < ∞, the space bpis reflexive. The space described in (2.1) is known as the K¨othe dual space of bp. In general, the K¨othe dual of a Banach sequence space does not coincide with its dual Banach space. The main result of [37] is the following isometric characterization of the dual space b∗ pof bp. Theorem 2.5 (Jagers).For u= (un)∞ n=1 ∈c0, adopting the convention u∞= B∞= 0, set (a) m(1) := max{k∈N∪{∞} :|uk|= maxj≥1|uj|}. (b) For n≥1, provided m(n)is defined and finite, m(n+ 1) := max {k∈N∪{∞} :k > m(n), |um(n)|−|uk| Bm(n)−Bk = min m(n)<j≤∞ |um(n)|−|uj| Bm(n)−Bj}, in other case m(n+ 1) is not defined. (c) D(u)is the set of all k≥1such that m(k)is defined and finite. (d) For p > 1, q∥u∥:=  ∑ n∈D(u)(|um(n)|−|um(n+1)| Bm(n)−Bm(n+1) )q(Bm(n)−Bm(n+1))  1/q , and for p= 1, ∞∥u∥:= sup n∈D(u) |um(n)|−|um(n+1)| Bm(n)−Bm(n+1) . 38 Chapter 2. The Ces`aro space of Dirichlet series Define qd:= {u∈c0:q∥u∥<∞}. Then b∗ pis isometrically isomorphic to qd which turns out to be a Banach space with norm q∥·∥. It must be noted that although Jagers considered real sequences, all the results remain valid for complex sequences, as (bp)R:= bp∩RNis a solid real Banach lattice of sequences and so bpis the solid complex Banach lattice bp= (bp)R+i(bp)R(see Section 1.4 in the Preliminaries). Observe that the exact calculation of q∥u∥can be quite involved. For the particular case of cesp, Bennett gave a simpler, though just isomorphic, identification of the dual space ces∗ p, [14, p.61]. For 1 < p < ∞, consider the Banach space d(p) consisting of all complex sequences u= (un)∞ n=1 satisfying ∥u∥d(p):= (∞ ∑ n=1 sup k≥n|uk|p)1/p <∞.(2.2) Given a sequence (un)∞ n=1, the sequence (˜un)∞ n=1 defined by ˜un:= sup k≥n|uk|, n ≥1, is known as the least decreasing majorant of (un)∞ n=1. Theorem 2.6 (Bennett).For 1<p<∞and 1/p + 1/q = 1, the dual space ces∗ pof cespcan be identified with the sequence space d(q). The relationship between their norms is given by 1 q∥u∥d(q)≤ ∥u∥ces∗ p≤(p−1)1/p∥u∥d(q).(2.3) Regarding equivalent expressions for the norm in cesp, the following is of interest. The Copson spaces cop(p), for 0 < p < ∞, defined by cop(p) := {x= (xk)∞ k=1 ∈CN:∞ ∑ n=1 (∞ ∑ k=n |xk| k)p <∞}, were studied by Bennett in [14]. When p≥1, they are Banach sequence spaces under the norm ∥x∥cop(p):= (∞ ∑ n=1 (∞ ∑ k=n |xk| k)p)1/p , x ∈cop(p). Bennett proved, for 1 <p<∞, that cop(p) and cespare actually the same set, and gave an equivalence for their norms, [14,§10 and Theorem 10.8]. Chapter 2. The Ces`aro space of Dirichlet series 39 Theorem 2.7 (Bennett).For 1< p < ∞we have that cesp=cop(p)and 1 p(p−1)1/p ∥x∥cop(p)≤ ∥x∥cesp≤p p−1∥x∥cop(p). Optimal values for the constants are, for 1< p ≤2, ∥x∥cop(p)≤(p−1)1/p ∥x∥cesp, and, for 2≤p < ∞, ∥x∥cesp≤ζ(p)1/p ∥x∥cop(p). The next equivalence, due to Bennett, [14, Theorem 11.5], is inspired by a celebrated inequality of Hilbert, [32, Chapter IX], which asserts that ∞ ∑ n=1 (∞ ∑ k=1 |xk| n+k−1)p ≤(π sin(π/p))p∞ ∑ k=1 |xk|p where 1 <p<∞, and the inequality is strict unless xn= 0 for all n. The space hil(p) is then defined to be the set of all sequences x= (xn)∞ n=1 for which ∥x∥hil(p)=(∞ ∑ n=1 (∞ ∑ k=1 |xk| n+k−1)p)1/p <∞. Theorem 2.8 (Bennett).For 1< p < ∞we have that hil(p) = cespand ∥x∥cesp≤ ∥x∥hil(p)≤π qsin(π/p)∥x∥cesp, where the constants are the best possible. Other properties of the standard basis (en)∞ n=1 were observed in [25, Proposition 2.1]. Proposition 2.9. The sequence of canonical vectors (en)∞ n=1 is an unconditional, boundedly complete and shrinking Schauder basis for cesp. As noted for the spaces bp, the Ces`aro space cespis a solid complex Banach lattice for the coordinate-wise order. In particular, it satisfies that given complex sequences a= (an)∞ n=1 and b= (bn)∞ n=1 such that |bn|≤|an|for all n≥1, if a∈cespthen b∈cespand ∥b∥cesp≤ ∥a∥cesp. Other important lattice property of cespis the order continuity: for every (am)∞ m=1 ⊂ces+ psuch that am↓0 it follows that ∥am∥cesp→0 as m→ ∞. This fact follows (in the same way as for ℓp) from applying the Dominated Convergence Theorem to the 40 Chapter 2. The Ces`aro space of Dirichlet series functions fmdefined on Nby fm(n) = (1 n∑n k=1 am k)p, where am= (am n)∞ n=1. Denoting by λthe counting measure on N, we have that fm∈L1(λ) and ∥fm∥L1(λ)=∥am∥p cespfor all m≥1. Also, we can apply directly the fact that ℓpis order continuous. Since cespis solid, its order continuity can be rewritten as: for every (am)∞ m=1 ⊂cesp∩RNsuch that am↑aand |am| ≤ bfor all m≥1 with b∈ces+ pit follows that a∈cespand am→ain cespas m→ ∞. 2.2 The Ces`aro space of Dirichlet series H(cesp) For 1 < p < ∞, the Ces`aro space of Dirichlet series H(cesp) consists of all Dirichlet series whose sequence of coefficients belongs to the Ces`aro space cesp, that is, H(cesp) := {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p <∞}. It is a linear space which becomes a Banach space of Dirichlet series when endowed with the norm ∥f∥H(cesp):= (∞ ∑ n=1 (1 n n ∑ k=1 |ak|)p)1/p , f ∈ H(cesp).(2.4) Note that 1 < p1≤p2<∞implies H(cesp1)⊆ H(cesp2). The space H(cesp) inherits many of its functional properties from the sequence space cesp, as they are isometrically isomorphic. Since the canonical vectors {em:m≥1}of cespcorrespond to the monomials {m−s:m≥1}in H(cesp), from Propositions 2.4 and 2.9 we have the following result. Proposition 2.10. The following statements hold: (a) For every f(s) = ∑∞ n=1 ann−s∈ H(cesp)it follows that the Dirichlet polynomials ∑N n=1 ann−sconverges to fin the norm of H(cesp). Moreover, from the monotony of the norm of cesp, ∥f∥H(cesp)= sup N∈N    N ∑ n=1 ann−s   H(cesp) . (b) The sequence of monomials {m−s:m≥1}is an unconditional, boundedly complete and shrinking Schauder basis for H(cesp). In particular, H(cesp)is reflexive. Chapter 2. The Ces`aro space of Dirichlet series 41 A further approximation for functions in H(cesp) is possible. Let (pk)∞ k=1 denote the sequence of the prime numbers written in increasing order. For r∈N, let Nr:= {n=pα1 1pα2 2. . . pαr r:α1, . . . , αr≥0}. Consider the map Qrdefined by f(s) = ∞ ∑ n=1 ann−s7→ Qr(f) := ∑ n∈Nr ann−s. The map Qris in fact a projection Qr:H(cesp)→ H(cesp). An remarkable property of the projection Qris its multiplicativity, namely, Qr(fg) = Qr(f)Qr(g), which holds for any pair of Dirichlet series fand g, see [45, p.157]. The order continuity of cespand the monotony of its norm give the following result. Proposition 2.11. For each f(s) = ∑∞ n=1 ann−sin H(cesp)it follows that ∑n∈Nrann−sconverges to fin the norm of H(cesp). Moreover, ∥f∥H(cesp)= sup r∈N   ∑ n∈Nr ann−s   H(cesp) . Since cespis a solid sequence space for the coordinate-wise order, H(cesp) is a solid space of Dirichlet series for the coefficient-wise order. We next show that H(cesp) is a Banach space of analytic functions. For this, we determine the abscissa of convergence and the abscissa of absolute convergence of H(cesp). Theorem 2.12. Every Dirichlet series f∈ H(cesp)converges, in fact absolutely, on the vertical half-plane C1/q, where qis the conjugate exponent of p. Moreover, the value 1/q cannot be improved, that is, σc(H(cesp)) = σa(H(cesp)) = 1/q. Consequently, H(cesp)is a Banach space of analytic functions on C1/q, which is a maximal domain. 48 Chapter 2. The Ces`aro space of Dirichlet series for k≤x≤k+ 1. Then ∫n m dx x2= n−1 ∑ k=m∫k+1 k g(x)dx ≤ n−1 ∑ k=m∫k+1 k((g(k+ 1) −g(k))(x−k) + g(k))dx = n−1 ∑ k=m 1 2(g(k+ 1) + g(k)) =1 2 n ∑ k=m+1 g(k) + 1 2 n−1 ∑ k=m g(k) = n−1 ∑ k=m+1 g(k) + 1 2(g(n) + g(m)) = n−1 ∑ k=m+1 1 k2+1 2(1 n2+1 m2)= n−1 ∑ k=m 1 k2+1 2(1 n2−1 m2). So, (2.13) holds. On the other hand, it is also true that ∫n m dx x2+1 2(1 m2−1 n2)=(1 m−1 n)(1 + 1 2m+1 2n), so (n−m mn )(1 + 1 2m+1 2n)≤B. Then, by (2.12), we can prove the validity of (2.10) if we show n+ 1 n2m(n−m)<(n−m mn )(1 + 1 2m+1 2n). But this last inequality can be rewritten as n+ 1 n<1 + 1 2m+1 2n, which is true since m<n. Thus, (2.8) holds and so, for every m∈N, the sequence (2.7) is strictly increasing. Thus, for each n∈Nwe have m(n) = n. So, Theorem 2.5 implies that D(u) = Nfor u= (n−s0)∞ n=1 and so ∥δs0∥= (n−s0) d(2) =(∞ ∑ n=1 n2(1 nσ−1 (n+ 1)σ)2)1/2 .(2.14) Chapter 2. The Ces`aro space of Dirichlet series 49 Next we estimate each of the terms in the previous sum. For this we write them as follows: n(1 nσ−1 (n+ 1)σ)=n·(n+ 1)σ−nσ nσ(n+ 1)σ =n (n+ 1)σ·((1 + 1 n)σ−1) =1 (n+ 1)σ·(1 + n−1)σ−1 n−1 =1 (n+ 1)σ·ψ(n−1), where ψ(x) = (1 + x)σ−1 x. Simple calculations show that ψ′(x)<0 for x > 0 and so ψdecreases in (0,∞), and ψ(0+) := lim x→0+ψ(x) = σ. Then, 2σ−1 = ψ(1) ≤ψ(n−1)≤ψ(0+) = σ and so 2σ−1 (n+ 1)σ≤n(1 nσ−1 (n+ 1)σ)≤σ (n+ 1)σ. Finally make the corresponding sums over all n∈Nto obtain. (2σ−1)√ζ(2σ)−1≤ ∥δs0∥ ≤ σ√ζ(2σ)−1. The following graph represents the bounds we have found for ∥δs0∥, depending on ℜ(s0), for point evaluations δs0on H(ces2). 50 Chapter 2. The Ces`aro space of Dirichlet series σ√ζ(2σ)−1 (2σ−1)√ζ(2σ)−1 √ζ(2σ) 1 2√ζ(2σ) √6/π √ζ(2)−1 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 1+ log ζ(2) log 2 bb We end this section exhibiting two equivalent expressions for the norm in H(ces2) which are of independent interest. Proposition 2.16. Let f(s) = ∑∞ n=1 ann−s∈ H(ces2). Consider the functionals M(f) := (∞ ∑ i,j=1 |ai||aj| max{i, j})1/2 , N(f) := (∞ ∑ n=1 |an| n n ∑ k=1 |ak|)1/2 . Then N(f)≤M(f)≤ ∥f∥H(ces2)≤√2M(f)≤2N(f). Chapter 2. The Ces`aro space of Dirichlet series 51 Proof. Note that (an)∞ n=1 ∈ces2. Then ∥f∥2 H(ces2)=∥(an)∞ n=1∥2 ces2 =∞ ∑ n=1 (1 n n ∑ k=1 |ak|)2 =∞ ∑ n=1 1 n2(∑ 1≤i,j≤n|ai||aj|) =∞ ∑ i,j=1 |ai||aj|(∑ n≥i,j 1 n2). Since 1/n ≤∑∞ k=nk−2≤2/n, for every n≥1, it follows that 1 max{i, j}≤∑ n≥i,j 1 n2≤2 max{i, j}. Hence, we deduce that M(f)≤ ∥f∥H(ces2)≤√2M(f). On the other hand, M(f)2=∞ ∑ i,j=1 |ai||aj| max{i, j} =∞ ∑ n=1 1 n(∑ max{i,j}=n|ai||aj|) =∞ ∑ n=1 |an| n(|an|+ 2 n−1 ∑ k=1 |ak|) ≤2∞ ∑ n=1 |an| n n ∑ k=1 |ak|= 2N(f)2. In a similar way M(f)2≥∞ ∑ n=1 |an| n n ∑ k=1 |ak|=N(f)2. Consequently, N(f)≤M(f)≤√2N(f). 52 Chapter 2. The Ces`aro space of Dirichlet series 2.3 The Banach spaces of Dirichlet series Ar, A2,r and H∞(Cr) There are other Banach spaces of Dirichlet series which will be needed in order to study the space H(cesp). The most important ones are the spaces Ar,A2,r, and H∞(Cr) for r∈R. We review them. For r∈R, the space Arconsists of all Dirichlet series whose sequence of coefficients belongs to the weighted space ℓ1((n−r)∞ n=1), that is, Ar:= {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 |an|n−r<∞}.(2.15) It is a solid Banach space under the coefficient-wise order and the norm ∥f∥Ar:= ∞ ∑ n=1 |an|n−r, f ∈ Ar. When r= 0, the corresponding space is the well known Wiener-Dirichlet algebra A+consisting of all Dirichlet series with absolutely summable coefficients, studied by Bayart, Finet, Li, and H. Queff´elec, see [13]. Proposition 2.17. The following statements hold: (a) Regarding the abscissae of convergence we have σc(Ar) = σa(Ar) = r. (b) For s0∈Cr, the point evaluation functional δs0is bounded on Arand ∥δs0∥= 1. Proof. (a) Note that σa(f)≤rfor all f∈ Arand thus σa(Ar)≤r. For arbitrary α > 1−r, we have that fα(s) := ∑∞ n=1 n−αn−s∈ Arand σc(Ar)≥σc(fα) = 1 −α. Making α→1−rit follows that σc(Ar)≥rand so the conclusion follows. (b) For s0∈Crand f(s) = ∑∞ n=1 ann−s∈ Ar, we have that |δs0(f)|=|f(s0)| ≤ ∞ ∑ n=1 |an| nℜ(s0)≤∞ ∑ n=1 |an| nr=∥f∥Ar. Hence δs0:Ar→Cis continuous with ∥δs0∥ ≤ 1. Actually ∥δs0∥= 1 as the function 1belongs to Arand has norm equal to one. Chapter 2. The Ces`aro space of Dirichlet series 53 In a similar way, for r∈R, the space A2,r consists of all Dirichlet series whose sequence of coefficients belongs to the weighted space ℓ2((n−2r)∞ n=1), that is, A2,r := {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 |an|2n−2r<∞}.(2.16) It is a solid Banach space under the coefficient-wise order and the norm ∥f∥A2,r := (∞ ∑ n=1 |an|2n−2r)1/2 , f ∈ A2,r. When r= 0, the corresponding space is the Hilbert space of Dirichlet series Hconsisting of all Dirichlet series with square summable coefficients, studied by Hedenmalm, Lindqvist, and K. Seip, see [35]. Proposition 2.18. The following statements hold: (a) Regarding the abscissae of convergence we have σc(A2,r) = σa(A2,r) = 1/2 + r. (b) For s0=σ+it ∈C1/2+r, the point evaluation functional δs0is bounded on A2,r and ∥δs0∥ ≤ √ζ(2(σ−r)). Proof. (a) Note, for every f∈ A2,r and σ > 1/2 + r, that ∞ ∑ n=1 |an| nσ≤(∞ ∑ n=1 1 n2(σ−r))1/2(∞ ∑ n=1 |an|2 n2r)1/2 =√ζ(2(σ−r)) ∥f∥A2,r ,(2.17) and so σa(f)≤1/2+r. Hence, σa(A2,r)≤1/2+r. For arbitrary α > 1/2−r, we have that fα(s) := ∑∞ n=1 n−αn−s∈ A2,r and σc(A2,r)≥σc(fα) = 1 −α. Making α→1/2−rit follows that σc(A2,r)≥1/2 + r. The result is thus established. (b) For each s0=σ+it ∈C1/2+rand f(s) = ∑∞ n=1 ann−s∈ A2,r, by (2.17) we have that |δs0(f)|=|f(s0)| ≤ ∞ ∑ n=1 |an| nσ≤√ζ(2(σ−r)) ∥f∥A2,r . Then δs0:A2,r →Cis continuous with ∥δs0∥ ≤ √ζ(2(σ−r)). 54 Chapter 2. The Ces`aro space of Dirichlet series Remark 2.19. The following continuous embeddings hold A1/2⊆ H(ces2)⊆ A2,1/2 Indeed, for f(s) = ∑∞ n=1 ann−s, since |an| ≤ n ∑ k=1 |ak| ≤ √n n ∑ k=1 |ak|k−1/2, considering the equivalent expression N(f) for the norm in H(ces2) given in Proposition 2.16 we have that ∞ ∑ n=1 |an|2 n≤N(f)2≤∞ ∑ n=1 |an| √n n ∑ k=1 |ak| √k≤(∞ ∑ n=1 |an| √n)2 . For r∈R, the space H∞(Cr) consists of all bounded analytic functions on Crwhich can be represented as a Dirichlet series in some vertical half-plane (actually in Crby Bohr’s Theorem 1.15), that is, H∞(Cr) = D∩H∞(Cr).(2.18) It is a linear space that will be endowed with the supremum norm ∥f∥H∞(Cr):= sup s∈Cr|f(s)|, f ∈ H∞(Cr). Note that these spaces are not solid for the coefficient-wise order. Proposition 2.20. The following statements hold: (a) Regarding the abscissae of convergence we have σc(H∞(Cr)) = rand σa(H∞(Cr)) = r+ 1/2. (b) For each s0∈Cr, the point evaluation functional δs0is bounded on H∞(Cr)and ∥δs0∥= 1. Proof. (a) As noted before if f∈ H∞(Cr) then σc(f)≤rand so σc(H∞(Cr)) ≤ r. For arbitrary α > 1−r, we have that fα(s) := ∑∞ n=1 n−αn−s∈ H∞(Cr) and σc(H∞(Cr)) ≥σc(fα) = 1 −α. Making α→1−rit follows that σc(H∞(Cr)) ≥r. Thus, σc(H∞(Cr)) = r. For the abscissa of absolute convergence, we first consider the case r= 0. It follows from results of Bohnenblust and Hille, and Balasubramanian, Calado and Queff´elec, see [9, Theorem 1.1.2)], that Chapter 2. The Ces`aro space of Dirichlet series 55 (i) ∑∞ n=1 |an|n−1/2<∞for all f(s) = ∑∞ n=1 ann−s∈ H∞(C0). (ii) For each 0 ≤α < 1/2 there exists fα(s) = ∑∞ n=1 ann−s∈ H∞(C0) such that ∑∞ n=1 |an|n−α=∞. These facts imply that σa(H∞(C0)) = 1/2. For r= 0 consider the translation map τr:D → D given by τr(f)(s) := f(s+r),(2.19) that is, τr(∞ ∑ n=1 ann−s)=∞ ∑ n=1 ann−(s+r)=∞ ∑ n=1 (ann−r)n−s. Note that σc(τr(f)) = σc(f)−rand the same holds for the abscissae of absolute and uniform convergence. The translation τrestablishes an isometric isomorphism between H∞(Cr) and H∞(C0), τr:H∞(Cr)→ H∞(C0), from which it follows that σa(H∞(Cr)) = r+σa(H∞(C0)) = r+ 1/2. Part (b) is obvious. The isometric isomorphism between H∞(Cr) and H∞(C0) allows a neat solution to establishing the completeness of H∞(Cr) for the supremum norm. The result of Hedenmalm, Lindqvist and Seip states that H∞(C0) is isometrically isomorphic to the M(H), the multiplier algebra of the Hilbert space of Dirichlet series H(Theorem 1.4). Since this last space is complete (for the operator norm) it follows that H∞(C0) is complete for the supremum norm. Hence, H∞(Cr) endowed with the supremum norm is a Banach space. The relation between the spaces Arand H∞(Cr) is clear: Ar⊆ H∞(Cr) (2.20) and the containment is continuous with embedding constant equal to one. Indeed, for f(s) = ∑∞ n=1 ann−sin Arand s=σ+it ∈Cr, we have |f(s)| ≤ ∞ ∑ n=1 |an|n−σ≤∞ ∑ n=1 |an|n−r=∥f∥Ar. However, that relation is in fact more precise. 56 Chapter 2. The Ces`aro space of Dirichlet series Proposition 2.21. Let f∈ H∞(Cr)with f(s) = ∑∞ n=1 ann−ssuch that an≥ 0for all n≥1. Then f∈ Arand ∥f∥H∞(Cr)=∥f∥Ar. Thus, the positive cones of Arand H∞(Cr)coincide. Moreover, the space Ar is the solid core of H∞(Cr). Proof. For α > r and N∈N, as an≥0 for all n≥1, we have ∥f∥H∞(Cr)≥ |f(α)|= ∞ ∑ n=1 ann−α≥ N ∑ n=1 ann−α. Making α→rit follows that ∑N n=1 ann−r≤ ∥f∥H∞(Cr). So, f∈ Ar, and ∥f∥Ar≤ ∥f∥H∞(Cr). Thus, both norms coincide. This shows that the positive cones of Arand H∞(Cr) coincide. In order to show that Aris the solid core of H∞(Cr), let Ebe a solid subspace of H∞(Cr). We have to show that E ⊆ Ar. Let f∈ E with f(s) = ∑∞ n=1 ann−sand denote by hthe Dirichlet series defined as h(s) := ∑∞ n=1 |an|n−s. Since Eis solid, we have that h∈ E ⊆ H∞(Cr). Then h∈ Ar and so f∈ Ar, as Aris solid. Consequently, E ⊆ Ar. Not only Dirichlet series in H∞(Cr) with nonnegative coefficients belong to Ar, but also some other series with real coefficients, that is, ∑∞ n=1 ann−sin H∞(Cr) such that an∈R, for all n≥1, whenever the sign of the coefficients behaves multiplicatively. Recall that sgn denotes the sign function. Theorem 2.22. Let f∈ H∞(Cr)with f(s) = ∑∞ n=1 ann−ssuch that an∈R for all n≥1, and sgn(aij) = sgn(ai)·sgn(aj), i, j ≥1.(2.21) Then f∈ Ar, and, moreover, ∥f∥Ar=∥f∥H∞(Cr). Proof. Recall that the translation operator τr, which maps finto f(·+r), establishes an isometric isomorphism between the spaces H∞(Cr) and H∞(C0). The operator τris also an isometric isomorphism between Arand A0. This fact, together with sgn(ann−r) = sgn(an), n ≥1, Chapter 2. The Ces`aro space of Dirichlet series 57 shows that in order to prove the result it suffices to establish the following: if f∈ H∞=H∞(C0) satisfies condition (2.21), then f∈ A0and ∥f∥A0= ∥f∥H∞. The result of Hedenmalm, Lindqvist and Seip states that M(H) = H∞ (Theorem 1.4). Thus, we can assume that f∈ M(H). We can also assume that an= 0 for some n≥1, as in other case f≡0 and there would be nothing to prove. Fix M∈Nsuch that aM= 0. Choose m∈N, and set b:= (bn)∞ n=1 with bn={sgn(an) if n|Mm 0 in other case. Note that if n|Mmthen bn= 0, as sgn(an)·sgn(aMm/n) = sgn(aMm) = sgn(aM)m= 0. Let g(s) := ∑∞ n=1 bnn−s. Thus, g∈ H, since it has a finite number of non-zero coordinates. Then, for card(I) denoting the cardinal of the set I, we have ∥fg∥H≤ ∥f∥M(H)∥g∥H=∥f∥M(H)(card({n:n|Mm}))1/2.(2.22) On the other hand, ∥fg∥2 H=∞ ∑ n=1 ∑ k|n bkan k 2 ≥∑ n∈{jM:j|Mm−1}∑ k|n bkan k 2 =∑ n∈{jM:j|Mm−1}∑ k|n sgn(ak)an k 2 , where for the last equality note that if kis a divisor of nwith n=jM and j|Mm−1, then kis a divisor of Mm, and so bk= sgn(ak). Next, appealing to 64 Chapter 2. The Ces`aro space of Dirichlet series (b) Let s0=σ+it ∈C1+1/q and f(s) = ∑∞ n=1 ann−s∈ H([C, ℓp]). By writing an=∑n k=1 ak−∑n−1 k=1 ak, for all n≥2, we obtain |f(s0)| ≤ ∞ ∑ n=1 |an|n−σ ≤ |a1|+∞ ∑ n=2 n−σ n ∑ k=1 ak+∞ ∑ n=2 n−σ n−1 ∑ k=1 ak ≤∞ ∑ n=1 n−σ n ∑ k=1 ak+∞ ∑ n=2 (n−1)−σ n−1 ∑ k=1 ak = 2 ∞ ∑ n=1 n−σ n ∑ k=1 ak = 2 ∞ ∑ n=1 n1−σ1 n n ∑ k=1 ak. So |f(s0)| ≤ 2(∞ ∑ n=1 1 np n ∑ k=1 ak p)1/p(∞ ∑ n=1 n(1−σ)q)1/q, that is, |f(s0)| ≤ 2∥f∥H([C,ℓp])ζ(q(σ−1))1/q. We conclude that ∥δs0∥ ≤ 2ζ(q(σ−1))1/q. Chapter 3 The multiplier algebra of H(cesp) In this chapter we look into the multiplier algebra M(H(cesp)) of H(cesp), that is, the space of all analytic functions f(on some domain containing C1/q) such that fg ∈ H(cesp) for all g∈ H(cesp). The chapter is organized in four sections. In the first section we review general results for the multiplier algebra M(E) of a space Eof Dirichlet series. The second section is devoted to the study of the multiplier algebra of the spaces Ar(for which M(Ar) = Ar) and A2,r (for which M(A2,r) = H∞(Cr)). We also consider the cases of M(H(ℓp)) and M(H([C, ℓp])). The third section is the main section of the chapter, where we study M(H(cesp)). We prove that monomials, and hence Dirichlet polynomials, are multipliers on H(cesp) and calculate their norm as multipliers (3.4). This allows to deduce that A1/q ⊆ M(H(cesp)) (Theorem 3.11 and Proposition 3.3). We prove, via point evaluations, that multipliers on H(cesp) are bounded functions on the vertical half-plane C1/q (Theorem 3.11 and Proposition 3.2). This last result allows to show that the spaces A1/q,M(H(cesp)) and H∞(C1/q) have the same positive cone and that A1/q is the solid core of M(H(cesp)), with respect to the coefficient-wise order (Proposition 3.13). A fine use of abscissae of convergence allows to deduce that M(H(cesp)) =H∞(C1/q) (Theorem 3.11). At this stage we know that the multiplier algebra M(H(cesp)) satisfies A1/q ⊆ M(H(cesp)) ⊊H∞(C1/q), and that A1/q is the solid core of M(H(cesp)). The fact that M(H(cesp)) = H∞(C1/q) shows a completely different situation to that of the multipliers algebras of the previously considered spaces of Dirichlet series, H,Hp,Hα,Ap µ, Bp,Dα. For these spaces the multiplier algebra is H∞. This fact is in accordance and actually follows from the classical result of Schur identifying the multiplier algebra of the Hardy space H2(D) with the space H∞(D), Theorem 65 66 Chapter 3. The multiplier algebra of H(cesp) 1.24. However, it is noteworthy that if we consider the space of all Taylor series on the unit disc Dwith coefficient belonging to cesp, it turns out that its multiplier algebra is not H∞(D) but a rather smaller algebra, namely, the Wiener algebra of all absolutely convergent Taylor series; [24, Theorem 3.1], [25, Theorem 4.1]. This motivates the conjecture M(H(cesp)) = A1/q, that is, multipliers are Dirichlet series f(s) = ∑∞ n=1 ann−ssatisfying the condition ∑∞ n=1 |an|n−1/q <∞. For establishing the conjecture we first prove it for multipliers f(s) = ∑∞ n=1 ann−ssuch that an= 0 only if the prime factorization of nhas primes no other than the first rprimes, p1, p2, . . . , pr, for a fixed r∈N(Theorem 3.15). The full identification is proven, with no restriction on the coefficients of the multipliers, in Theorem 3.17. In the last section we study the existence of compact multipliers (Theorem 3.20) and the sequence multipliers from H(cesp) into its multiplier algebra M(H(cesp)), that is, we study how “close” is H(cesp) from M(H(cesp)) (Theorem 3.21). The chapter is written in a somewhat inefficient way, at least from the viewpoint of conciseness. The reason for this choice is the following. The effort and time devoted to proving the identification of the multiplier algebra of H(cesp) was largely augmented by the fact that most of the time we were led by a wrong conjecture and not pursuing the right result. It was, however, that effort and the detailed analysis that it required what eventually led to the correct result. This is presented in Theorem 3.17. The previous results which are presented correspond to the steps that helped formulating the right conjecture. 3.1 General facts on multiplier algebras We collect several facts on the multipliers and multiplier algebra of a Banach space of Dirichlet series. Although some of them are known, we include the proofs for the sake of completeness. Given a Banach space of Dirichlet series E ⊆ D with convergence abscissa σc(E), a multiplier on Eis an analytic function fon Cσc(E)with the property that fg ∈ E for every g∈ E. The multiplier algebra of Eis the space of all multipliers on E, which will be denoted by M(E). Proposition 3.1. Let E ⊆ D be a Banach space of Dirichlet series. Suppose that there exists σ≥σc(E)such that the point evaluation functional δs0is continuous on Efor every s0∈Cσ. Then the following holds: Chapter 3. The multiplier algebra of H(cesp) 67 (a) For every f∈ M(E), the operator Mf:E → E, given by g∈ E 7→ Mf(g) := fg ∈ E, is linear and bounded. (b) If the constant function 1∈ E, then M(E)⊆ E and for every f∈ M(E) it follows that ∥f∥E≤ ∥1∥E∥f∥M(E),(3.1) where ∥f∥M(E)denotes the operator norm of Mf. Moreover, in this case: (b.1) M(E)is a closed subspace of the space B(E)of all bounded linear operators of Einto itself. (b.2) Endowing M(E)with the norm ∥·∥M(E), the containment M(E)⊆ Eis continuous with embedding constant equal to ∥1∥E. (b.3) For every f∈ M(M(E)), the operator Mf:M(E)→ M(E)defines a bounded linear operator and, with equality of norms, we have M(M(E)) = M(E). Proof. (a) We use the closed graph theorem. Let f∈ M(E) and suppose that gn→0 and Mf(gn)→hin E. For every s0∈Cσ, since δs0is continuous on E, it follows that gn(s0)→0 and f(s0)gn(s0) = Mf(gn)(s0)→h(s0). Then h= 0 on Cσand, by Theorem 1.13,h= 0. Consequently, the operator Mfis bounded. (b) Suppose that 1∈ E. Then, for f∈ M(E) we have that f=Mf(1)∈ E and ∥f∥E=∥Mf(1)∥E≤ ∥1∥E∥f∥M(E). (b.1) Let (fn)∞ n=1 ⊆ M(E) be such that Mfn→T∈B(E) in the operator norm. Then, for each g∈ E we have that fn·g→T(g) in Eand so fn(s0)g(s0)→T(g)(s0) for all s0∈Cσ. On the other hand, since (fn)∞ n=1 is a Cauchy sequence in Eby (3.1), there exists f∈ E such that fn→fin E and so pointwise on Cσ. Then, T(g)(s0) = f(s0)g(s0) for s0∈Cσand thus T(g) = f·g. Hence, f∈ M(E) and T=Mf. (b.2) is clear from the previous results. (b.3) Since M(E)⊆ E continuously we have that the point evaluation δs0 is continuous on M(E) for every s0∈Cσ. Noting that σ≥σc(E)≥σc(M(E)), by (a), every f∈ M(M(E)) defines a bounded linear operator Mfon M(E). Since 1∈ M(E), by (b.2) we have that M(M(E)) ⊆ M(E) and ∥f∥M(E)≤ ∥f∥M(M(E)) 68 Chapter 3. The multiplier algebra of H(cesp) for all f∈ M(M(E)), as ∥1∥M(E)= 1. Conversely, let f∈ M(E). For every g∈ M(E) and h∈ E, as gh ∈ E, we have that fgh ∈ E and ∥fgh∥E≤ ∥f∥M(E)∥gh∥E≤ ∥f∥M(E)∥g∥M(E)∥h∥E. Then, fg ∈ M(E) with ∥fg∥M(E)≤ ∥f∥M(E)∥g∥M(E). Thus, f∈ M(M(E)) with ∥f∥M(M(E)) ≤ ∥f∥M(E). Next proposition shows that, under minimal conditions which guarantee a good behavior of M(E), every multiplier on Eis bounded on the aproppriate domain. Proposition 3.2. Let E ⊆ D be a Banach space of Dirichlet series satisfying the condition of Proposition 3.1 for some σ≥σc(E)and such that 1∈ E. Then, M(E)⊆ H∞(Cσ), where the containment is continuous with continuity constant equal to one. Proof. Let f∈ M(E). By Proposition 3.1, we have that M(E)⊆ E and so f2=ff ∈ E with ∥f2∥E≤ ∥f∥E∥f∥M(E)≤ ∥1∥E∥f∥2 M(E). Iterating the above procedure, we obtain, for every n≥1, that fn∈ E and ∥fn∥E≤ ∥1∥E∥f∥n M(E). For each s0∈Cσ, by hypothesis, the point evaluation functional δs0is bounded on E. Then |fn(s0)|=|δs0(fn)| ≤ ∥δs0∥·∥fn∥E ≤ ∥δs0∥·∥1∥E∥f∥n M(E). Since |fn(s0)|=|f(s0)|n, it follows that |f(s0)| ≤ (∥δs0∥·∥1∥E)1/n∥f∥M(E). Making n→ ∞ we have that |f(s0)| ≤ ∥f∥M(E). Hence, f∈ H∞(Cσ) and ∥f∥H∞(Cσ)≤ ∥f∥M(E). Moreover, if the monomials n−s, for n≥1, are multipliers on E, a weighted ℓ1space of Dirichlet series is contained in M(E). Chapter 3. The multiplier algebra of H(cesp) 69 Proposition 3.3. Let E ⊆ D be a Banach space of Dirichlet series satisfying the condition of Proposition 3.1 for some σ≥σc(E)and such that 1∈ E. If {n−s:n≥1} ⊂ M(E), denoting µn=∥n−s∥M(E), we have that A((µn)∞ n=1) := {f(s) = ∞ ∑ n=1 ann−s:∞ ∑ n=1 |an|µn<∞}⊆ M(E) and ∥f∥M(E)≤∑∞ n=1 |an|µnfor all f∈ A((µn)∞ n=1). Proof. Let f(s) = ∑∞ n=1 ann−s∈ A((µn)∞ n=1). The series ∑∞ n=1 ann−sis absolutely convergent in M(E), as ∞ ∑ n=1 ∥ann−s∥M(E)=∞ ∑ n=1 |an|µn<∞, and so it converges in norm to some h∈ M(E). Since, M(E)⊆ E continuously and so norm convergence in M(E) implies pointwise convergence on Cσ, it follows that f=h∈ M(E). From the equality above it follows that ∥f∥M(E)≤ ∑∞ n=1 |an|µn. Under the natural conditions of Proposition 3.1, a multiplier on Eis in fact a Dirichlet series in E. The pointwise product f(s)g(s) of two Dirichlet series f(s) = ∑∞ n=1 ann−sand g(s) = ∑∞ n=1 bnn−sis, in the appropriate vertical halfplane in which both series converge, the Dirichlet series h(s) = ∑∞ n=1 cnn−s whose sequence of coefficients c= (cn)∞ n=1 is given by the Dirichlet convolution c:= a·bof the sequences a= (an)∞ n=1 and b= (bn)∞ n=1, that is, cn= (a·b)n:= ∑ k|n akbn k=∑ k|n an kbk, n ≥1, where k|ndenotes that kis a divisor of n. Remark 3.4. There is a matrix representation for the action on Eof every Dirichlet series belonging to M(E). Given f(s) = ∑∞ n=1 ann−s, consider the infinite matrix Af= (αij)∞ i,j=1 ={αij =ai/j if j|i, 0 in other case, 70 Chapter 3. The multiplier algebra of H(cesp) that is, Af=                        a100000000000··· a2a10 0 0 0 0 0 0 0 0 0 ··· a30a1000000000··· a4a20a10 0 0 0 0 0 0 0 ··· a50 0 0 a10 0 0 0 0 0 0 ··· a6a3a20 0 a10 0 0 0 0 0 ··· a70 0 0 0 0 a10 0 0 0 0 ··· a8a40a20 0 0 a10 0 0 0 ··· a90a30 0 0 0 0 a10 0 0 ··· a10 a50 0 a20 0 0 0 a10 0 ··· a11 000000000a10··· a12 a6a4a30a20 0 0 0 0 a1··· . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ....                        . Note, for every b= (bn)∞ n=1 ∈CN, that Afb=(∞ ∑ k=1 αnkbk)∞ n=1 =(∑ k|n an kbk)∞ n=1 and, for g(s) = ∑∞ n=1 bnn−s, Mf(g)(s) = f(s)g(s) = ∞ ∑ n=1 (∑ k|n an kbk)n−s. Considering the sequence space b E:= {(bn)∞ n=1 ∈CN:∞ ∑ n=1 bnn−s∈ E}, it follows that f∈ M(E), that is Mf:E → E, if and only if Af:b E → b E. Remark 3.5. Consider the monomials {m−s:m≥1}associated to the canonical vectors {em:m≥1}. For a Dirichlet series g(s) = ∑∞ n=1 bnn−s, setting b= (bn)∞ n=1, the coefficients of the Dirichlet series m−sgare given by the sequence em·b. For each n∈Nwe have (em·b)n=∑ k|n em kbn k={bn mif m|n, 0 in other case, ={biif n=im for some i≥1, 0 in other case, Chapter 3. The multiplier algebra of H(cesp) 71 that is, em·b= (0, ..., 0, b1 |{z} m ,0, ..., 0, b2 |{z} 2m ,0, ..., 0, b3 |{z} 3m , ...), Hence, multiplying a Dirichlet series by the monomial m−s“spreads out” by intervals of msteps the coefficients of the initial Dirichlet series. 3.2 First examples of multiplier algebras The multiplier algebras of the spaces of Dirichlet series Ar,A2,r and H∞(Cr) studied in Section 2.3 can be identified. The situation is much more complicated for the spaces H(ℓp) and H([C, ℓp]) studied in Section 2.4, nevertheless, although we have no identification for their algebras, we show some results related to them. Let us start with the space Ar, which turns out to coincide isometrically with its multiplier algebra. Proposition 3.6. The equality M(Ar) = Arholds with equal norms. Proof. From Propositions 2.17 and 3.1, it follows that M(Ar)⊆ Arcontinuously with embedding constant equal to one. For the converse containment consider the monomials {m−s:m≥1}. Given g(s) = ∑∞ n=1 bnn−s∈ Ar, setting b= (bn)∞ n=1, we have that em·bis the sequence of coefficients of the Dirichlet series m−sg(see Remark 3.5). Then, since ∞ ∑ n=1 |(em·b)n|n−r=∞ ∑ i=1 |(em·b)im|(im)−r=m−r∞ ∑ i=1 |bi|i−r=m−r∥g∥Ar, it follows that m−sg∈ Arand ∥m−sg∥Ar=m−r∥g∥Ar. Hence, m−s∈ M(Ar) and ∥m−s∥M(Ar)≤m−r. Actually ∥m−s∥M(Ar)=m−r, as 1∈ Arwith norm one and ∥m−s∥Ar=m−r. Therefore, from Proposition 3.3, the containment Ar⊆ M(Ar) holds continuously with embedding constant equal to one. The space H∞(Cr) also coincides isometrically with its multiplier algebra, as can be directly checked, but moreover it coincides with the multiplier algebra of the space A2,r. Proposition 3.7. The equality M(A2,r) = H∞(Cr)holds with equal norms. Morever, Ar⊊M(A2,r) = H∞(Cr)⊊A2,r, where the containments are continuous with continuity constant equal to one. 72 Chapter 3. The multiplier algebra of H(cesp) Proof. From Propositions 2.18 and 3.1 it follows that M(A2,r)⊆ A2,r continuously with embedding constant equal to one. Consider the translation map τr:D → D given by τr(f)(s) = f(s+r) = ∞ ∑ n=1 (ann−r)n−s for f(s) = ∑∞ n=1 ann−s, (2.19). Note that τr(fg) = τr(f)τr(f) for all f, g ∈ D. We have observed before that τr:H∞(Cr)→ H∞(C0) is an isometric isomorphism. Also, τrclearly induces an isometric isomorphism τr:A2,r → A2,0=H. Let us show that M(A2,r) = H∞(Cr) by using translation map τrand the fact that M(H) = H∞(C0) with equality of norms (Theorem 1.4). Let f∈ H∞(Cr) and g∈ A2,r. Since τr(f)∈ H∞(C0) = M(H) and τr(g)∈ H, we have that τr(fg) = τr(f)τr(g)∈ H and so fg ∈ A2,r. Moreover, ∥fg∥A2,r =∥τr(fg)∥H =∥τr(f)τr(g)∥H ≤ ∥τr(f)∥M(H)∥τr(g)∥H =∥τr(f)∥H∞(C0)∥g∥A2,r =∥f∥H∞(Cr)∥g∥A2,r . Then f∈ M(A2,r) with ∥f∥M(A2,r)≤ ∥f∥H∞(Cr). Let now f∈ M(A2,r) and g∈ H. Since τ−r(g)∈ A2,r we have that fτ−r(g)∈ A2,r and so τr(f)g=τr(fτ−r(g)) ∈ H. Moreover, ∥τr(f)g∥H=∥τr(fτ−r(g))∥H =∥fτ−r(g)∥A2,r ≤ ∥f∥M(A2,r)∥τ−r(g)∥A2,r =∥f∥M(A2,r)∥g∥H. Then τr(f)∈ M(H) = H∞(C0), that is, f∈ H∞(Cr), and ∥f∥H∞(Cr)=∥τr(f)∥H∞(C0)=∥τr(f)∥M(H)≤ ∥f∥M(A2,r). The strictness of the inclusion H∞(Cr)⊊A2,r follows from Propositions 2.18 and 2.20 which state that σc(A2,r) = 1 2+rand σc(H∞(Cr)) = r. Chapter 3. The multiplier algebra of H(cesp) 73 We have seen in Section 2.3 that Ar⊆ H∞(Cr) continuously with continuity constant equal to one, (2.20). Since, by Propositions 2.17 and 2.20, we have that σa(Ar) = rand σa(H∞(Cr)) = r+1 2, it follows that Ar⊊H∞(Cr). Consider now the space H(ℓp). From Propositions 2.23 and 3.1 we have that M(H(ℓp)) ⊆ H(ℓp) continuously with embedding constant equal to one. The next proposition shows other inclusion relations. Proposition 3.8. For qbeing the conjugate exponent of p, the containments A0⊆ M(H(ℓp)) ⊊H∞(C1/q) hold continuously with continuity constant equal to one. Proof. From Proposition 3.2 it follows that M(H(ℓp)) ⊆ H∞(C1/q) continuously with continuity constant equal to one. The inclusion is strict, as σa(M(H(ℓp))) ≤σa(H(ℓp)) = 1/q < 1/q + 1/2 = σa(H∞(C1/q)), see Propositions 2.23 and 2.20. For the first containment, let us consider the monomials {m−s:m≥1}. Given g(s) = ∑∞ n=1 bnn−s∈ H(ℓp), the coefficients of the Dirichlet series m−sg are given by the sequence em·b, where b= (bn)∞ n=1 (see Remark 3.5). Then, since ∞ ∑ n=1 |(em·b)n|p=∞ ∑ i=1 |bi|p=∥g∥p H(ℓp), it follows that m−sg∈ H(ℓp) and ∥m−sg∥H(ℓp)=∥g∥H(ℓp). Hence, m−s∈ M(H(ℓp)) and ∥m−s∥M(H(ℓp)) ≤1. Actually ∥m−s∥M(H(ℓp)) = 1, as 1∈ H(ℓp) with norm one and ∥m−s∥H(ℓp)= 1. Therefore, from Proposition 3.3, the containment A0⊆ M(H(ℓp)) holds continuously with embedding constant equal to one. We finish this section by looking at the multiplier algebra of H([C, ℓp]). Note that 1∈ H([C, ℓp]) with ∥1∥H([C,ℓp]) =ζ(p)1/p. From Propositions 2.24 and 3.1 we have that M(H([C, ℓp])) ⊆ H([C, ℓp]) continuously with embedding constant ζ(p)1/p. Further inclusion relations follow in a similar way as for the multiplier algebra of H(ℓp). Proposition 3.9. For qbeing the conjugate exponent of p, the containments A1/q ⊆ M(H([C, ℓp])) ⊊H∞(C1+1/q) hold continuously with continuity constants equal to one. 80 Chapter 3. The multiplier algebra of H(cesp) where Kis a constant, depending solely on pand r, obtained via the mean value theorem applied to the function (1 + x)p: (1 + Br Arnα)p≤1 + Br Arnαmax {p(1 + x)p−1: 0 < x < Br Arnα} ≤1 + 1 nα(Br Ar p(1 + Br Ar)p−1) = 1 + K nα. Thus, ∥gα∥p H(cesp)≤Ap r αp(ζ(p(1 −α)) + Kζ(p(1 −α) + α)).(3.8) On other hand, observe that each k∈Ncan be written as k=ω γ with ω∈Nrand γ∈Ir. Then (a·bα)k=∑ j|k ajbα k j =∑ j|k j∈Nr ajbα k j =aωbα γ.(3.9) Indeed, if j∈Nris such that j|k, it must then be that j|ω. Moreover, if j=ω we have that k/j ∈ Irand so bα k/j = 0. Fix m∈Nand consider the set Nm r:= {n= r ∏ i=1 pαi i: 0 ≤αi≤mfor all 1 ≤i≤r}. Since ωγ = ˆωˆγ, with ω, ˆω∈Nrand γ, ˆγ∈Ir, implies that ω= ˆωand γ= ˆγ, it follows that the set O=∪ ω∈Nm r ω·Ir Chapter 3. The multiplier algebra of H(cesp) 81 is a finite union of disjoint sets. Then, ∥fgα∥p H(cesp)=∥a·bα∥p cesp=∞ ∑ n=1 1 np(n ∑ k=1 |(a·bα)k|)p ≥∞ ∑ n=prm r 1 np(n ∑ k=1 k∈O |(a·bα)k|)p =∞ ∑ n=prm r 1 np(∑ ω∈Nm r n ∑ k=1 k∈ω·Ir |(a·bα)k|)p =∞ ∑ n=prm r 1 np(∑ ω∈Nm r ⌊n/ω⌋ ∑ γ=1 γ∈Ir |(a·bα)ω γ|)p =∞ ∑ n=prm r 1 np(∑ ω∈Nm r |aω|⌊n/ω⌋ ∑ γ=1 γ∈Ir |bα γ|)p =∞ ∑ n=prm r 1 np(∑ ω∈Nm r |aω|⌊n/ω⌋ ∑ γ=1 γ∈Ir 1 γ1−α)p. Note that ω≤prm rwhenever ω∈ Pm r, and so n/ω ≥1 for n≥prm r. In order to apply Lemma 3.16 we restrict to n≥n0:= 24qr+2q(m−1)prm r. In this case, for every ω∈Nm r, we have Brωα Ar =2r−1ωα ∏r i=1 (1−1 pi) ≤2r−1ωα ∏r i=1 1 2 = 22r−1ωα ≤22r+(m−1) 2mpαrm r ≤24qrα+2qα(m−1) 2mpαrm r ≤nα 2m, since 1 <2qα, as we are assuming that 1/(2q)< α < 1/q. Consequently 1−Brωα Arnα≥1−1 2m 82 Chapter 3. The multiplier algebra of H(cesp) and, from Lemma 3.16, we have ⌊n/ω⌋ ∑ γ=1 γ∈Ir 1 γ1−α≥Ar(⌊n ω⌋+ 1)α α−Br α ≥Arnα αωα(1−Brωα Arnα) ≥Arnα αωα(1−1 2m). Then ∥fgα∥p H(cesp)≥∞ ∑ n=n0 1 np(∑ ω∈Nm r |aω|Arnα αωα(1−1 2m))p ≥(1−1 2m)pAp r αp(∞ ∑ n=n0 1 np(1−α))(∑ ω∈Nm r |aω| ωα)p ,(3.10) ≥(1−1 2m)pAp r αp(p(1 −α)−1)np(1−α)−1 0(∑ ω∈Nm r |aω| ωα)p . From (3.8) and (3.10) we have ∥f∥M(H(cesp)) ≥∥fgα∥H(cesp) ∥gα∥H(cesp) =(1−1 2m)(∑ ω∈Nm r |aω| ωα)M(α)1/p,(3.11) for M(α)−1:=(p(1 −α)−1)(24qr+2q(m−1)prm r)p(1−α)−1 ·(ζ(p(1 −α)) + Kζ(p(1 −α) + α)). Inequality (3.11) holds for every αwith 1/(2q)< α < 1/q. Thus, we can take limit when α→(1/q)−. Since p(1 −α)→p(1 −1/q) = 1, it follows that lim α→(1/q)−M(α) = lim α→(1/q)−(p(1 −α)−1)ζ(p(1 −α)) = 1. Consequently, ∥f∥M(H(cesp)) ≥lim α→(1/q)−(1−1 2m)(∑ ω∈Nm r |aω| ωα)M(α)1/p =(1−1 2m)(∑ ω∈Nm r |aω| ω1/q ). Chapter 3. The multiplier algebra of H(cesp) 83 Since the above inequality holds for arbitrary m∈N, it follows that ∥f∥M(H(cesp)) ≥∑ ω∈Nr |aω| ω1/q . Proof of Lemma 3.16.We fix all throughout the proof 0 < α < 1. We first establish (3.6) in the case when r, n ∈Nsatisfy n≤pr. In this case, Ir∩{1,2, .., pr}={1}, so we have n ∑ k=1 k∈Ir 1 k1−α= 1. The inequalities α≤Br≤Arnα+Brestablish the right-hand side of (3.6). For the left-hand side, direct calculation shows, when r= 1, that Ar(pr+ 1)α−Br≤α. For r≥2 we have Ar(n+ 1)α−Br≤(1−1 2)(1−1 3)(pr+ 1)α−2r−1 ≤pr+ 1 3−2r−1 ≤2r+ 1 3−2r−1 =1 3(1 −2r−1)<0, where we use that pr≤2r. Hence, (3.6) holds when n≤pr. Next we consider the case when n≥pr+ 1. We will prove it by induction on r. For r= 1 and n≥p1+ 1 = 3 we have to show that (n+ 1)α 2−1≤α n ∑ k=1 k∈I1 1 k1−α≤nα 2+ 1.(3.12) Observe that I1={k∈N: 2 .k}={k∈N:kis odd}. Then n ∑ k=1 k∈I1 1 k1−α= n ∑ k=1 1 k1−α− n ∑ k=1 keven 1 k1−α = n ∑ k=1 1 k1−α−1 21−α ⌊n/2⌋ ∑ j=1 1 j1−α. 84 Chapter 3. The multiplier algebra of H(cesp) Since (n+ 1)α−1≤α n ∑ k=1 1/k1−α≤nα, n ∈N, it follows that α n ∑ k=1 k∈I1 1 k1−α≤nα−1 21−α((⌊n 2⌋+ 1)α−1) ≤nα−1 21−α(nα 2α−1) =nα 2+1 21−α ≤nα 2+ 1. In a similar way, α n ∑ k=1 k∈I1 1 k1−α≥((n+ 1)α−1)−1 21−α⌊n 2⌋α ≥((n+ 1)α−1)−1 21−α nα 2α =((n+ 1)α−1)−nα 2 ≥((n+ 1)α−1)−(n+ 1)α 2 =(n+ 1)α 2−1. So (3.12) holds. Consequently, (3.6) holds for r= 1 and all n≥1. For the inductive step, assume that (3.6) holds for a certain r∈Nand all n≥1, we will prove that (3.6) holds for r+ 1 and all n≥pr+1 + 1. Observe that Ir+1 ={k∈N:pi.kfor all 1 ≤i≤r+ 1} ={k∈N:pi.kfor all 1 ≤i≤r}\{k∈N:pr+1|k} =Ir\({k∈N:pr+1|k}∩Ir), Chapter 3. The multiplier algebra of H(cesp) 85 and {k∈N:pr+1|k}∩Ir={jpr+1 :j∈Ir}. It follows that n ∑ k=1 k∈Ir+1 1 k1−α= n ∑ k=1 k∈Ir 1 k1−α− n ∑ k=1 k∈{jpr+1:j∈Ir} 1 k1−α = n ∑ k=1 k∈Ir 1 k1−α−1 p1−α r+1 ⌊n/pr+1⌋ ∑ j=1 j∈Ir 1 j1−α. By the inductive hypothesis, (3.6) holds for rand all n≥1. Thus α n ∑ k=1 k∈Ir+1 1 k1−α≤Arnα+Br−1 p1−α r+1 (Ar(⌊n pr+1 ⌋+ 1)α−Br) ≤Arnα+Br−1 p1−α r+1 (Ar nα pα r+1 −Br) =Arnα(1−1 pr+1 )+Br(1 + 1 p1−α r+1 ) ≤Arnα(1−1 pr+1 )+ 2Br =Ar+1nα+Br+1. On the other hand, α n ∑ k=1 k∈Ir+1 1 k1−α≥Ar(n+ 1)α−Br−1 p1−α r+1 (Ar⌊n pr+1 ⌋α+Br) ≥Ar(n+ 1)α−Br−1 p1−α r+1 (Arnα pα r+1 +Br) ≥Ar(n+ 1)α−Br−1 p1−α r+1 (Ar(n+ 1)α pα r+1 +Br) =Ar(n+ 1)α(1−1 pr+1 )−Br(1 + 1 p1−α r+1 ) ≥Ar(n+ 1)α(1−1 pr+1 )−2Br. =Ar+1(n+ 1)α−Br+1. Therefore is (3.6) established. Why is it not possible to deduce that M(H(cesp)) = A1/q from the conclusion of Theorem 3.15, despite the fact that the result holds for all r∈N and the bound obtained∑ n∈Nr|an|n−1/q ≤ ∥f∥M(H(cesp)) 86 Chapter 3. The multiplier algebra of H(cesp) is independent of r? The reason, or at least one of the reasons, lays on the properties of the multiplier algebra M(H(cesp)) with respect to the coefficient-wise order. Consider a multiplier f∈ M(H(cesp)) with f(s) = ∑∞ n=1 ann−s. For r∈N, let Qr(f) be the Dirichlet series whose coefficients are given by Qr(f)n:= {ann∈Nr 0n∈ Nr. The Dirichlet series Qr(f) satisfies the condition placed on the coefficients in Theorem 3.15, but, since we do not know that the multiplier algebra M(H(cesp)) is solid for the coefficient-wise order, we cannot guarantee that Qr(f) is a multiplier on H(cesp). Even if we could prove, for all r∈N, that Qr(f)∈ M(H(cesp)) and so, Qr(f)∈ A1/q with ∥Qr(f)∥A1/q =∥Qr(f)∥M(H(cesp)), we could neither guarantee that supr∥Qr(f)∥M(H(cesp)) <∞. This is why from Theorem 3.15 we cannot conclude that conjecture (3.5) is true. Note, however, that if M(H(cesp)) were solid, then by Remark 3.14 we would immediately have the conjecture (3.5). It turns out that it is more successful to carefully analyze the proof of Theorem 3.15, with the aim of removing the restriction that an= 0 for n∈ Nr for some r∈N. The key point of the proof of Theorem 3.15 is that the definition of the sequence bα(3.7), allows the critical fact, (3.9), that k∈Nimplies k=ω γ, ω ∈Nr, γ ∈Ir, and (a·bα)k=aωbα γ. A different definition of the sequence b, (3.14), allows a related critical fact, (3.16), (a·b)k=aωh(r), which leads to the identification of the multiplier space. Theorem 3.17. For 1< p < ∞and 1/p + 1/q = 1, we have M(H(cesp)) = A1/q with equality of norms. Chapter 3. The multiplier algebra of H(cesp) 87 Proof. Let f∈ M(H(cesp)) with f(s) = ∑∞ n=1 ann−s. Recall that prdenotes the r-th prime number. Since by the Prime Number Theorem lim r→∞ pr rlog r= 1, for each m∈Nthere exists rm∈Nwith rm> m such that 1−1 m≤pr rlog r≤1 + 1 m,for all r≥rm, and so (1−1 m)rlog r≤pr≤(1 + 1 m)rlog r, for all r≥rm.(3.13) For a fixed m∈N, consider the sequence b= (bn)∞ n=1 defined by bn:= {h(r)n=prwith r≥rm, 0 in other case, (3.14) for an appropriate positive function hto be determined later. Let g(s) := ∑∞ n=1 bnn−s. Then, ∥g∥p H(cesp)=∥b∥p cesp=∞ ∑ n=1 1 np(n ∑ k=1 |bk|)p =∞ ∑ n=prm 1 np(∞ ∑ r=rm pr≤n h(r))p.(3.15) Note that for each k=ω prwith ω∈ Pmand r≥rm, it follows that (a·b)k=∑ j|k ajbk j=aωbpr=aωh(r).(3.16) Indeed, if j|kand j=ωwe have that k/j =prfor all r≥rmand so bk/j = 0. For the value m∈Nfixed above, consider the subset of Ngiven by Pm:= {rm−1 ∏ i=1 pαi i: 0 ≤αi≤mfor all 1 ≤i≤rm−1}. Since ωpr= ˆωpˆrwith ω, ˆω∈ Pmand r, ˆr≥rmimplies that ω= ˆωand pr=pˆr, it follows that the set Om=∪ ω∈Pm ω·{pr:r≥rm} 88 Chapter 3. The multiplier algebra of H(cesp) is a finite union of disjoint sets. Then, for any nm≥pmrm+1 rm, we have that ∥fg∥p H(cesp)=∥a·b∥p cesp=∞ ∑ n=1 1 np(n ∑ k=1 |(a·b)k|)p ≥∞ ∑ n=nm 1 np(n ∑ k=1 k∈Om |(a·b)k|)p =∞ ∑ n=nm 1 np(∑ ω∈Pm ∞ ∑ r=rm pr≤n ω |(a·b)ω pr|)p =∞ ∑ n=nm 1 np(∑ ω∈Pm|aω|∞ ∑ r=rm pr≤n ω h(r))p.(3.17) Note that ω≤pmrm rmwhenever ω∈ Pmand so n/ω ≥prmfor n≥nm. We use the next lemma which will be proved afterwards. Set 1/2q < α < 1/q and ϕ(x) := (xlog x)α, x ∈[1,∞). Note that the derivative ϕ′of ϕis positive. Define h:= ϕ′. Lemma 3.18. There exists xqlarge enough such that (βm m+ 1)α−(rmlog rm)α≤∑ r=rm pr≤β h(r)≤(βm m−1)α(3.18) whenever rm≥xqand β≥prm. Note that (βm m+1 )α−(rmlog rm)α≥0precisely when rmlog rm(1 + 1 m)≤β. Assume that rm≥xq. From (3.15) and Lemma 3.18 we have that ∥g∥p H(cesp)=∥b∥p cesp≤∞ ∑ n=prm 1 np(nm m−1)pα =(m m−1)pα ∞ ∑ n=prm 1 np(1−α)(3.19) ≤(m m−1)pα 1 (p(1 −α)−1)(prm−1)(p(1−α)−1) . Chapter 3. The multiplier algebra of H(cesp) 89 On the other hand, for nm= 2r2 mpmrm+2q rm, from (3.17) and Lemma 3.18 it follows that ∥fg∥p H(cesp)=∥a·b∥p cesp ≥∞ ∑ n=nm 1 np(∑ ω∈Pm|aω|((nm ω(m+ 1))α−(rmlog rm)α))p . Note, for n≥nmand ω∈ Pm, that (nm ω(m+ 1))α−(rmlog rm)α=(nm ω(m+ 1))α(1−(ω(m+ 1)rmlog rm nm )α) ≥(nm ω(m+ 1))α(1−1 prm), since ω(m+ 1)rmlog rm m≤pmrm rm(m+ 1)rmlog rm m ≤2r2 mpmrm+2q rm p2q rm =nm p2q rm ≤nm p1/α rm ≤n p1/α rm . Then, ∥fg∥p H(cesp)=∥a·b∥p cesp ≥∞ ∑ n=nm 1 np(∑ ω∈Pm|aω|(nm ω(m+ 1))α(1−1 prm))p =(1−1 prm)p(m m+ 1)pα ∑ n≥nm 1 np(1−α)(∑ ω∈Pm |aω| ωα)p (3.20) ≥(1−1 prm)p(m m+ 1)pα 1 (p(1 −α)−1)n(p(1−α)−1) m(∑ ω∈Pm |aω| ωα)p . 96 Chapter 3. The multiplier algebra of H(cesp) Proof. For N≥1, set AN:= |a1|+. . . +|aN|and A0:= 0. Abel’s summation formula gives N ∑ n=1  anφ(n) n1/q = N ∑ n=1 (An−An−1)|φ(n)| n1/q =AN|φ(N)| N1/q + N−1 ∑ n=1 An(|φ(n)| n1/q −|φ(n+ 1)| (n+ 1)1/q ). Since φis differentiable, |φ(n)| n1/q −|φ(n+ 1)| (n+ 1)1/q ≤ φ(n) n1/q −φ(n+ 1) (n+ 1)1/q  ≤max n≤x≤n+1  d dx(φ(x)x−1/q) = max n≤x≤n+1 φ′(x)x−1/q −1 qφ(x)x−1/q−1 ≤max n≤x≤n+1 n−1/q φ′(x)−φ(x) qx  Thus, N ∑ n=1  anφ(n) n1/q ≤AN|φ(N)| N1/q + N ∑ n=1 An n1/q ·max n≤x≤n+1 φ′(x)−φ(x) qx . Recall that, according to Proposition 2.2, the growth of the sequence a∈ cespis restrained by lim n→∞ 1 n1/q n ∑ k=1 |ak|= 0. It follows that the sequence (Ann−1/q)∞ n=1 is bounded by a constant K > 0, and ANφ(N)N−1/q →0 as N→ ∞, since φis bounded. Then ∞ ∑ n=1 |anφ(n)| n1/q ≤K∞ ∑ n=1 max n≤x≤n+1 φ′(x)−φ(x) qx <∞. Example 3.22. For φ(x) := log−αx, for α > 1 with α > 1 considered on [2,∞), we have max n≤x≤n+1 φ′(x)−φ(x) qx ≤α+ 1/q nlogαn, n ≥2. Chapter 3. The multiplier algebra of H(cesp) 97 Thus, we are in the conditions of Proposition 3.21, and so for every ∑∞ n=1 ann−s∈ H(cesp) we have that ∞ ∑ n=2 an logαnn−s is a multiplier on H(cesp). Remark 3.23. Proposition 3.21 can alternatively be viewed as follows: under condition (3.24) the sequence (φ(n))∞ n=1 is a multiplier sequence from H(cesp) into A1/q, in the sense that ∞ ∑ n=1 anφ(n)n−s∈ A1/q for each ∞ ∑ n=1 ann−s∈ H(cesp). Chapter 4 The Ces`aro operator acting on Dirichlet series We end this memoire with a brief comment concerning the Ces`aro averaging operator Con sequences, a= (an)∞ n=1 ∈CN7−→ C(a) := (1 n n ∑ k=1 ak)∞ n=1 ∈CN. The aim is to study the Ces`aro operator when acting on different spaces of Dirichlet series via their sequence of coefficients, f(s) = ∞ ∑ n=1 ann−s7−→ C(f)(s) := ∞ ∑ n=1 (1 n n ∑ k=1 ak)n−s, s ∈C.(4.1) For example, Hardy’s inequality for p= 2, ∞ ∑ n=1 (1 n n ∑ k=1 |ak|)2 <4∞ ∑ n=1 |an|2, can be interpreted as asserting that Cmaps Hinto Hboundedly, ∥C(f)∥H≤2∥f∥H, f ∈ H. In fact, the construction of the sequence space ces2guarantees that Cmaps H(ces2) into Hboundedly, ∥C(f)∥H≤ ∥f∥H(ces2), f ∈ H(ces2). What happens, for example, for the spaces Hp, 1 ≤p < ∞, defined by Bayart, in the case when p= 2? For studying this last question (and 99 100 Chapter 4. The Ces`aro operator acting on Dirichlet series other similar ones) we present an integral formula for the action of the Ces`aro operator on Dirichlet series analogous to the classical integral formula available for the action of the Ces`aro operator on Taylor series, see (4.2) below. The deduction of such formula is shown in (4.5) and (4.6) below. Unfortunately, the study was not continued further. Formula for the Ces`aro operator acting on Taylor series The Ces`aro operator acting on the space H(D) of analytic functions on the open unit disk Dis C:H(D)→H(D), defined by f(z) = ∞ ∑ n=0 anzn7−→ C(f)(z) := ∞ ∑ n=0 (1 n+ 1 n ∑ k=0 ak)zn, z ∈D. Indeed, if the Taylor series for fdefines an analytic function on D, then the Taylor series for C(f) also defines an analytic function on D. For f(z) = ∑∞ n=0 anzn∈H(D) and z0∈D, we have that ∞ ∑ n=0 (1 n+ 1 n ∑ k=0 ak)zn 0=∞ ∑ k=0 ak ∞ ∑ n=k zn 0 n+ 1 =1 z0 ∞ ∑ k=0 ak ∞ ∑ n=k zn+1 0 n+ 1 =1 z0 ∞ ∑ k=0 ak ∞ ∑ n=k∫z0 0 ξndξ =1 z0∫z0 0 ∞ ∑ k=0 ak ∞ ∑ n=k ξndξ =1 z0∫z0 0 1 1−ξ ∞ ∑ k=0 akξkdξ. Then, we arrive to the integral expression C(f)(z0) = 1 z0∫z0 0 f(ξ) 1−ξdξ, z0∈D,(4.2) which has many applications in the study of the Ces`aro operator. Formula for the Ces`aro operator on Dirichlet series We look for an integral formula similar to (4.2) for the Ces`aro operator acting on Dirichlet series in the sense of (4.1). First note, for σ > max{σa(f),0}, Chapter 4. The Ces`aro operator acting on Dirichlet series 101 that ∞ ∑ n=1  1 n n ∑ k=1 akn−σ≤∞ ∑ n=1 1 n1+σ n ∑ k=1 |ak| =∞ ∑ k=1 |ak|∞ ∑ n=k 1 n1+σ ≤ |a1|(1 + 1 σ)+1 σ ∞ ∑ k=2 |ak| (k−1)σ ≤ |a1|(1 + 1 σ)+2σ σ ∞ ∑ k=2 |ak| kσ<∞. Then, σa(C(f)) ≤max{σa(f),0}. In particular C:D → D, as σa(f)≤1 + σc(f) (Theorem 1.12). Following a suggestion by Prof. Juan Arias de Reyna, we will use the Perron-Landau formulae: for c, x > 0 1 2πi ∫c+i∞ c−i∞ xs sds =   0 if x < 1, 1/2 if x= 1, 1 if x > 1, where for x= 1 the integral is a Cauchy principal value, see [36, Theorem G, p. 75]. For a Dirichlet series f(s) = ∑∞ n=1 ann−s∈ D,c > max{σa(f),0}and x > 0, the Perron-Landau formula gives ∑ n<x an=1 2πi ∫c+i∞ c−i∞ f(s)xs sds (4.3) when x∈ Nand ∑ n<x an+ax 2=1 2πi p.v. ∫c+i∞ c−i∞ f(s)xs sds (4.4) when x∈N. Then, via formula (4.3), for s0∈Ccand 0 < r < 1, we have that ∞ ∑ n=1 (1 n n ∑ k=1 ak)n−s0=∞ ∑ n=1 1 n1+s0∑ k<n+r ak =1 2πi ∞ ∑ n=1 1 n1+s0∫c+i∞ c−i∞ f(s)(n+r)s sds =1 2πi ∫c+i∞ c−i∞ f(s)(∞ ∑ n=1 (n+r)s ns0+1 )ds s. 102 Chapter 4. The Ces`aro operator acting on Dirichlet series We arrive to C(f)(s0) = 1 2πi ∫c+i∞ c−i∞ f(s)(∞ ∑ n=1 (n+r)s ns0+1 )ds s(4.5) whenever ℜ(s0)> c > max{σa(f),0}and 0 < r < 1. We can follow the same steps using (4.4) instead of (4.3), to obtain, for s0∈Cc, ∞ ∑ n=1 (1 n n ∑ k=1 ak)n−s0=∞ ∑ n=1 1 n1+s0(∑ k<n ak+an 2)+1 2 ∞ ∑ n=1 an n1+s0 =1 2πi ∞ ∑ n=1 1 n1+s0p.v. ∫c+i∞ c−i∞ f(s)ns sds +1 2 ∞ ∑ n=1 an n1+s0 =1 2πi p.v. ∫c+i∞ c−i∞ f(s)(∞ ∑ n=1 ns ns0+1 )ds s+1 2 ∞ ∑ n=1 an n1+s0. 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