Defining small-scale biorefineries based on a multi-factorial statistical typology Authors: Aicha Ait Sair, Bernard Cathala, Franck Michaud, and Kamal Kansou Abstract:

The concept of biorefinery is one of the pillars of the bioeconomy, which promotes the diversification of transformation paths of bioresources. It is in fact defined as the sustainable conversion of biomass into a multiple bio-based products. In that, it is a broad concept that encompasses a multitude of technologies, modalities and scales. Generally, the scale of a biorefinery is brought to its processing or production capacities. Other attempts linked the scale of a biorefinery to its economic viability. However, to distinguish small-scale biorefineries (SSB) from the large one remain challenging. In this paper, we propose a definition of the SSB based on a multi-factorial analysis using statistical approaches. The methodological approach consisted of collecting data on 15 operational biorefineries as case studies, the selection of the variables (investment cost, processing capacity, process complexity, mobility of the process, biomass type and added value) and their statistical analysis using factorial analysis of mixed data (FAMD) and hierarchical ascending clustering (HAC). It resulted in the clustering of biorefineries generating four clusters: The smallest scale, the small scale, the hybrid and the large scale. From that, we extracted that the small scale can be defined by a small investment cost (less than 2M€), a small processing capacity (less than 100t/day), with a low process complexity able to produce from low to high added value products. The mobility of the facilities is also a crucial element in defining the small scale, since it ensure the proximity to biomass providers, but built-on site plants respecting the above characteristics can be small-scale as well. In addition, we found that the scale of a biorefinery depend on a trade-off between the mobility, the process and the socio-economic context. This equilibrium is necessary for the biorefinery economic and sustainable viability. It is even more essential in the case of small-scale biorefineries, to encounter the effects of not benefitting from the economy of scale.

Aim and approach used:

The aim of this work is to define the concept of a small-scale a biorefinery based on statistical analysis of existing biorefineries. In this work, we introduce an approach that identifies different-scale based designs of biorefineries, by clustering the biorefineries according to the various factors that define them: feedstock, process, economy and mobility of the facility.

In a first step, case studies of biorefineries auto-defining as small scale or farm scale were collected from various websites and biorefinery platforms. The focus was mainly on biorefineries that processed lignocellulosic biomass, or wood waste, and produced biomaterials and biochemicals. Energy and fuel producing biorefineries, which are usually regarded as large scale, were added to the list for representation purposes. 15 operational case studies were selected based on two principles: no missing data and a coverage of distinct variables. The selected variables were based on the literature and on expert-opinion: Processing capacity, investment cost, mobility of the biorefinery, process complexity, added value, and biomass type.

Initially, the dataset collected was subject to an analysis by two variables: Processing capacity and Investment Cost. It resulted in a non-discriminative dispersion, which showed that the two factors were not enough to dissect a tendency between scales. To that effect, the dataset was recoded to fit for clustering analysis. A combination of multifactorial data analysis and hierarchical clustering analysis were applied to, respectively, visualize the similarities between individuals and study their proximities and relevance [START_REF] Centr Pagès | Analyse factorielle de données mixtes[END_REF], and to build a hierarchy of clusters by merging pairs of clusters as the hierarchy moves up.

Scientific innovation and relevance:

The hypothesis made for this work is that the scale of biorefineries can be outlined by studying existing biorefineries, and analyzing the various factors that define them. It was based on the fact that, in the literature, there were no objective criteria that discriminated the large from the small scale biorefineries (Aristizábal-Marulanda and Cardona Alzate, 2019). Generally, when the scale is brought up, it is linked to the production or processing volume [START_REF] Serna Loaiza | Small-Scale Biorefineries : Future and Perspectives[END_REF]. An attempt of defining the scale adopted the economic viability as the single criterion to define the Minimum Processing Scale for the Economic Feasibility (Serna-Loaiza et al., 2019). However, biorefineries are complex entities that intertwine several elements in their constitution: different feedstock, types of processes, different products and uses, etc. [START_REF] Cherubini | Toward a common classification approach for biorefinery systems[END_REF]. In addition to the technological features, biorefineries depend on the surrounding socio-economic and environmental context, which underlines that defining a scale-based design for biorefineries require taking into consideration all the potential characteristics.

In that sense, by this work, we intend to bring a potential response to defining the scale-based design of biorefineries based on core aspects, and distinguishing by that, the small-scale biorefineries from the large ones.

Results and conclusions:

The statistical analysis of the factors defining the case studies resulted in four clusters identified as: smallest scale, small scale, hybrid and large scale biorefineries. This distinction came out of the variable rates and modalities of the investment cost, the processing capacity, the mobility of the biorefinery, the process complexity and the added value. These factors not only identified the scale-based designs of biorefineries, but they revealed that a trade-off is crucial between these factors, in order for biorefineries to sustain and be viable. The following figure (Fig. 1) represents the radar chart of the scale-based biorefineries based on the factors that define them.

Fig. 1

 1 Fig.1 Radar charts of the biorefinery scale-based design classes. Note that Biorefinery Mobility has only two qualitative modalities: "Mob", for mobile, and "Centr" for centralized.


F R Éd

Éric Bayart
email: frederic.bayart@uca.fr

Maofa Wang

Xingxing Yao

TOPOLOGICAL STRUCTURE OF THE SPACE OF COMPOSITION OPERATORS ON THE HARDY SPACE OF DIRICHLET SERIES

Keywords: Mathematics Subject Classification. Primary 47B33, Secondary 30B50, 46E15 Composition operator, Dirichlet series, topological structure, compactness, linear combination

The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.

Introduction

Let H be a Hilbert space of analytic functions on a domain U of the complex plane C and let ϕ be an analytic selfmap of U. The composition operator C ϕ is the linear operator defined by

C ϕ f = f • ϕ, f ∈ H. A central
problem in the investigation of composition operators is to relate function theoretic properties of the symbol ϕ to operator theoretic properties of C ϕ , see the books [START_REF] Cowen | Composition operators on spaces of analytic functions[END_REF][START_REF] Shapiro | Composition operators and classical function theory[END_REF][START_REF] Zhu | Operator theory in function spaces[END_REF].

In recent decades, the study of the topological structure, for the operator norm topology, of the set C(H) of composition operators acting on H attracted the attention of many mathematicians. The earliest work is Berkson's [START_REF] Berkson | Composition operators isolated in the uniform operator topology[END_REF] isolation theorem in the setting of Hardy-Hilbert space H 2 (D) on the unit disk D in C, and then this result was generalized by MacCluer [START_REF] Maccluer | Components in the space of composition operators[END_REF]. The understanding of the topological structure of the set of composition operators was linked to the problem of determining if the difference of two composition operators is compact by the Shapiro and Sundberg conjecture raised in [START_REF] Shapiro | Isolation amongst the composition operators[END_REF]: C ϕ 0 , C ϕ 1 belong to the same connected component of C(H 2 (D)) if and only if C ϕ 0 -C ϕ 1 is compact. Bourdon [START_REF] Bourdon | Components of linear fractional composition operators[END_REF] and independently Moorhouse and Toews [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF] disproved this conjecture but they also showed clearly why these problems are closely related.

Beyond compact differences, the compactness of linear combinations of composition operators has also attracted the attention of many mathematicians. As a sample of results it is known that composition operators induced by linear fractional maps of the unit ball B n in C n behave quite rigidly in the following sense: the compactness of linear combinations N j=1 λ j C ϕ j , for finitely many distinct linear fractional maps ϕ j and nonzero complex numbers λ j , implies that each composition operator C ϕ j is compact on the Hardy space H 2 (B n ) (see [START_REF] Choe | Compact linear combinations of composition operators induced by linear fractional maps[END_REF]). It should be noticed that the study of the compactness of linear combinations of composition operators is a more difficult problem than the study of the compactness of linear differences, since high order cancelations may occur, and that this often requires new ideas. For example, see [START_REF] Choe | Compact double differences of composition operators on the Bergman spaces[END_REF][START_REF] Choe | Compact linear combination of composition operators on Bergman spaces[END_REF][START_REF] Kriete | Linear relations in the Calkin algebra for composition operators[END_REF] and references therein.

Our aim, in the present work, is to make a thorough study of these problems on the Hardy-Hilbert space H of Dirichlet series. Hedenmalm-Lindqvist-Seip [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF] introduced and started the study of the Hilbert space of Dirichlet series with square summable coefficients:

H = f (s) = ∞ n=1 a n n -s : f = ∞ n=1 |a n | 2 1/2 < ∞ .
By the Cauchy-Schwarz inequality, the functions in H are all analytic on the halfplane C 1/2 (where, for θ ∈ R, C θ = {s ∈ C : Re s > θ} and C + = C 0 ). A characteristic feature of the space H is its reproducing kernel function being essentially the Riemann zeta function: K w (s) = ζ(w + s) for w, s ∈ C 1/2 , where ζ(z) = ∞ n=1 n -z . The main results in [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coefficients[END_REF][START_REF] Queffélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF] show that an analytic map ϕ : C 1/2 → C 1/2 induces a bounded composition operator C ϕ on H if and only if it is a member of the Gordon-Hedenmalm class G defined as follows.

Definition. The Gordon-Hedenmalm class G consists of the maps ϕ :

C 1/2 → C 1/2 of the form ϕ(s) = c 0 s + ψ(s),
where c 0 is a non-negative integer (called the characteristic of ϕ, i.e., char (ϕ) = c 0 ), and ψ(s) = ∞ n=1 c n n -s converges uniformly in C ε for every ε > 0 and has the following properties: From then on, several authors have studied the properties of composition operators acting on H or on similar spaces of Dirichlet series, see for instance [START_REF] Bailleul | Composition operators on weighted Bergman spaces of Dirichlet series[END_REF][START_REF] Bailleul | Composition operators on Bohr-Bergman spaces of Dirichlet series[END_REF][START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF][START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF][START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF][START_REF] Brevig | Norms of composition operators on the H 2 space of Dirichlet series[END_REF][START_REF] Queffélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF]. In particular, the study of the compactness of composition operators on H is a difficult problem. Very recently, when the symbol has zero characteristic, Brevig and Perfekt [START_REF] Brevig | A mean counting function for Dirichlet series and compact composition operators[END_REF] gave a characterization of compactness by introducing a mean-counting function which has to satisfy a decay condition. Nevertheless, a similar characterization is unknown for symbols with positive characteristic, and even for simple symbols with zero characteristic, subtle phenomena occur.

Let us introduce some terminology. A symbol ϕ ∈ G is said to have unrestricted range if inf

s∈C 0 Re ϕ(s) = 1/2 if c 0 = 0, 0 if c 0 ≥ 1.
Correspondingly, if ϕ(C 0 ) is strictly contained in any smaller half-plane, we say that ϕ has restricted range. It is well-known that the composition operator C ϕ is compact on H when ϕ has restricted range [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF]Theorem 21]. We call ϕ a Dirichlet polynomial symbol, if

ϕ(s) = c 0 s + c 1 + n k=2 c k k -s .
In this work, we will be interested in polynomial symbols satisfying some conditions on their degree. Recall the definition of the arithmetic function Ω(n

) := α 1 +• • •+α d provided n = p α 1 1 • • • p α d
d where P = (p i ) i≥1 denotes the set of prime numbers. If ψ(s) = n k=1 c n k -s is a Dirichlet polynomial, its degree is the infimum of the integers N such that c n = 0 provided Ω(n) > N .

We will also be interested in composition operators C ϕ generated by linear symbols

ϕ(s) = c 0 s + c 1 + d k=1 c q k q -s k ,
where {q k } d k=1 is a sequence of finitely multiplicatively independent positive integers, and q k = 1, c q k = 0 for each k = 1, • • • , d. It should be observed that a linear symbol is not necessary a polynomial symbol of degree less than or equal to 1. Following [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF] and [START_REF] Finet | Compactness of composition operators on a Hilbert space of Dirichlet series[END_REF], it is known that the composition operator C ϕ induced by the linear symbol ϕ with c 0 = 0 is compact if and only if ϕ has restricted range or d ≥ 2; when c 0 ≥ 1, C ϕ is compact if and only if ϕ has restricted range. For a Dirichlet polynomial symbol ϕ with c 0 ≥ 1, Bayart-Brevig [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF] showed that C ϕ is compact if and only if ϕ has restricted range.

Let us come back to our initial problems of the topological structure of C(H) and of the compactness of linear combinations of composition operators on H. The first result in this direction was obtained by the authors in [START_REF] Bayart | Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series[END_REF]: if ϕ 1 and ϕ 2 are linear symbols and λ 1 , λ 2 are nonzero complex numbers, then λ 1 C ϕ 1 + λ 2 C ϕ 2 is compact if and only if both C ϕ 1 and C ϕ 2 are compact. We extend this result in two directions, allowing any finite linear symbols, and allowing polynomial symbols of degree at most 2 (see Theorems 4.5,4.6 and 4.8).

To prove this, we will develop a machinery, inspired by [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF], allowing to get lower bounds for the essential norm of a linear combination of composition operators. Many problems arise in the setting of H: the lack of reproducing kernels defined on C + , the infinite polydisc D ∞ hidden behind C + , the characteristic of the symbol which forces to adopt different proofs following its value. For instance, we will show that a basic result of Shapiro and Sundberg (the set of compact composition operators on H 2 (D) is connected) does not extend to C(H), but that the set of compact composition operators with prescribed characteristic is connected.

Most of these results are necessary conditions for the difference of two composition operators to be compact (or for two operators to be in the same connected component). In Section 5 we turn to sufficient conditions. We succeed (see Theorem 5.5 for instance) to give sufficient conditions that are easy to testify to prove that C ϕ 0 -C ϕ 1 is compact, or that C ϕ 0 and C ϕ 1 belong to the same connected component. These results involve sufficient conditions of independent interest to prove that some weighted composition operators are bounded or compact. Again the cases of zero and nonzero characteristics require quite different approaches, one based on Carleson measures, one based on Nevanlinna counting functions. Finally we mention that in some cases, the necessary and sufficient conditions coincide, giving a characterization of the compactness of C ϕ 0 -C ϕ 1 .

Throughout this paper we use the notation X Y or Y X for non-negative functions X and Y to mean that there exists C > 0 such that X ≤ CY , where C does not depend on the associated variables. Similarly, we use the notation X ≈ Y if both X Y and Y X hold. We will sometimes write X a Y if we want to point out that the involved constant depend on a. Given two positive functions f (x) and g(x) defined on (a, +∞), we write f (x)

∼ x→a g(x) (respectively, f (x) ∼ x→+∞ g(x)), provided lim x→a f (x) g(x) = 1 (respectively, lim x→+∞ f (x) g(x) = 1).

Preliminaries

This section first introduces some background material associated to Bohr lifts, vertical limit functions, reproducing and partial reproducing kernels.

2.1. Bohr lifts. At some places, it will be useful to work on the polydisc

D d instead of C 0 . Let f (s) = ∞
n=1 a n n -s be a Dirichlet series and let Q = {q 1 , . . . , q d } be a sequence of multiplicatively independent integers such that for each n with a n = 0, we have n = q

α := q α 1 1 • • • q α d d with α ∈ N d 0 .
We will say that f has d multiplicative generators when d < +∞, and that Q is compatible with f . The associated Bohr lift

B Q f (z) is the holomorphic function defined on D d by B Q f (z) = ∞ n=1 a n z α 1 1 • • • z α d d .
By Kronecker's theorem (see for example [START_REF] Queffélec | Diophantine approximation and Dirichlet series[END_REF]), the map R → T d , t → (q it 1 , . . . , q it d ) has dense range. In particular, if

f maps C 0 into C 0 , then B Q f maps D d into C 0 , and f (iR) = B Q f (T d ).
For any Dirichlet series f having finitely multiplicative generators, for all d sufficiently large, the sequence of the first d prime numbers {p 1 , . . . , p d } is always compatible with f . This Bohr lift will be simply denoted by B. It induces a unitary map from H onto the Hardy space H 2 (D ∞ ) over the infinite polydisc D ∞ . It should be observed that the degree of a Dirichlet polynomial f is exactly the degree of its Bohr lift Bf . 2.2. Vertical limit functions. Following [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF], let Ξ be the dual group of Q + , where Q + denotes the multiplicative discrete group of strictly positive rational numbers. Consequently, Ξ is the set of all functions χ :

Q + → C, such that (a) χ(mn) = χ(m)χ(n) for all m, n in Q + . (b) |χ(n)| = 1.
Ξ can be identified with T ∞ , the Cartesian product of countably many copies of the unit circle. Indeed, given any point z = (z 1 , z 2 , . . .) ∈ T ∞ , we define the value of χ at the primes through

χ(2) = z 1 , χ(3) = z 2 , . . . , χ(p m ) = z m , . . .
and extend the definition multiplicatively. This then yields a character, and clearly all characters can be obtained in this procedure. Also the unique normalized Haar measure on Ξ can be identified with the ordinary product measure m on T ∞ . For f (s) = ∞ n=1 a n n -s ∈ H and χ ∈ T ∞ , the vertical limit function f χ is defined by

f χ (s) := ∞ n=1 a n χ(n)n -s .
Note that the vertical translate function T τ f (s) := f (s + iτ ), τ ∈ R, corresponds to f χ with χ(n) = n -iτ . The name vertical limit function is justified by [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF]Lemma 2.4], which asserts that the functions f χ are precisely those obtained from the Dirichlet series f by taking a limit of vertical translations,

f χ (s) = lim k→∞ T τ k f (s).
Here the convergence is uniform on compact subsets of the half-plane where f converges uniformly. The vertical limit function f χ sometimes has better properties than the original function f . As explained in [20, Section 4.2], if f is in H, the Dirichlet series f χ converges in C 0 for almost every χ ∈ T ∞ , and the non-tangential boundary value f * χ (it) := lim σ→0 + f χ (σ + it) exists for almost every χ ∈ T ∞ and almost all t ∈ R. By [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF]Theorem 4.1], we can compute the norm of f in terms of the function f χ : if µ is a finite Borel measure on R, then

f 2 µ(R) = T ∞ R |f χ (it)| 2 dµ(t)dm(χ).
(2.1)

For any symbol ϕ(s) = c 0 s + ψ(s), define ϕ χ as ϕ χ (s) = c 0 s + ψ χ (s) for χ ∈ T ∞ , a vertical limit of the functions ϕ τ (s) = c 0 s + ψ(s + iτ ). As is clarified in [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coefficients[END_REF], for any such symbol ϕ : C 1/2 → C 1/2 , and f ∈ H, χ ∈ T ∞ , the following relation holds:

(f • ϕ) χ (s) = f χ c 0 • ϕ χ (s), s ∈ C 1/2 .
Especially, for any Dirichlet series ϕ ∈ G and f ∈ H, C ϕ f converges uniformly in C ε for every ε > 0, and (C ϕ f ) χ (s) = C ϕχ f χ c 0 (s). This implies that C ϕχ f = C ϕ f , and thus ϕ χ belongs to G for every χ ∈ T ∞ .

2.3.

Reproducing and partial reproducing kernels. As recalled in the introduction, the reproducing kernel at w

∈ C 1/2 in H is defined by K w (s) = ζ(w + s).
Reproducing kernels are often useful in the study of composition operators (especially in the study of their compactness) since C * ϕ (K w ) = K ϕ(w) . Moreover, we know that

K w 2 = ζ(2Re w) ∼ Re w→1/2 1 2Re w -1 and that if {w k } is any sequence of C 1/2 such that Re w k → 1/2, then K w k / K w k tends weakly to 0.
It turns out that in the specific case of composition operators acting on H, the reproducing kernels K w have a serious drawback: they are limited to C 1/2 whereas the symbol ϕ may be extended to C 0 . This motivates the introduction of partial reproducing kernels (refer to [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF] for more details). For w ∈ C 0 , and d ≥ 1, the partial reproducing kernel of order d at w is defined by

K d,w (s) = d j=1 n≥1 p -n(w+s) j = d j=1 1 1 -p -(w+s) j = n≥1, p + (n)≤p d n -w n -s
where p + (n) denotes the biggest prime factor of n. The function K d,w reproduces partially

H: if f (s) = ∞ n=1 a n n -s ∈ H, then f, K d,w = n≥1, p + (n)≤p d a n n -w .
By Euler's identity, we have

K d,w = d j=1 1 1 -p -2Re w j 1/2 =: ζ d (2Re w) 1/2 .
Moreover, it is clear that k d,w k := K d,w k / K d,w k converges weakly to zero for any sequence {w k } ⊆ C 0 satisfying Re w k → 0 and that

K d,w 2 ∼ Re w→0 A d (Re w) d
for some positive constant A d depending only on d.

The interest of using partial reproducing kernels in the study of composition operators comes from [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]Proposition 5]

: if ϕ(s) = c 0 s+ ∞ n=1 c n n -s belongs to G with c n = 0 for p + (n) > p d , then C * ϕ K d,w = K d,ϕ (
w) when c 0 ≥ 1, and C * ϕ K d,w = K ϕ(w) when c 0 = 0. We will see later (Lemma 3.6) how to generalize this formula to any symbol.

We shall also use a variant of these partial reproducing kernels. For an integer q ≥ 2 and w ∈ C 0 , we set K (q) w (s) = (1 -q -w q -s ) -1 , s ∈ C 1/2 . Clearly, K (q) w ∼ C q (Re w) -1/2 as Re w → 0 and, setting k

(q) w = K (q) w / K (q)
w , any sequence (k (q) w k ) converges weakly to zero in H as soon as Re w k goes to zero. The following lemma comes from [START_REF] Bayart | Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series[END_REF].

Lemma 2.1. Let r ≥ 2 be an integer and ϕ(s) = c 1 + c r r -s ∈ G.

(i) Assume that r = q n for some positive integer n. Then C * ϕ (K

w ) = K ϕ(w) . (ii) Assume that r m = q n for all positive integers n and m. Then C * ϕ (K

(q) w ) = K c 1 .
3. Connected components in the space of composition operators 3.1. The role of the characteristic. It is a result of Shapiro and Sundberg [START_REF] Shapiro | Isolation amongst the composition operators[END_REF] that the collection of compact composition operators on H 2 (D) is arcwise connected. This result breaks down for composition operators on H because of the characteristic: two composition operators with different characteristics cannot belong to the same component.

Theorem 3.1. The map C(H) → N, C ϕ → char (ϕ) is continuous.

We prove this result by a reasoning on the coefficients of p -ϕ(s) which can be adapted to a much more general context than H. We need to introduce the following definition.

Definition 3.2. Let l ≥ 1. A weighted partition of l is a triple (r, i, γ) with r ≥ 1, i ∈ {1, . . . , l} r satisfying i 1 < i 2 < • • • < i r , γ ∈ N r and γ 1 i 1 + • • • + γ r i r = l.
The set of weighted partitions of l will be denoted by WP l . Lemma 3.3. Let l ≥ 1 and let p be a prime number. For each (r, i, γ) a weighted partition of l, there exists a real number u(p, l, r, i, γ) such that, for all sequences c = (c k ) k≥2 with σ a ( k c k k -s ) < +∞, for all prime numbers q,

p -k≥2 c k k -s = +∞ k=1 α k (c)k -s with α q l (c) = -c q l log p + (r,i,γ)∈WP l ir<l u(p, l, r, i, γ)(c q i 1 ) γ 1 • • • (c q ir ) γr .
Proof. We can write formally

p -k≥2 c k k -s = k≥2 +∞ n=0 (-1) n (c k ) n n! (log p) n (k n ) -s .
If we expand the infinite product and since q is prime, the term (q l ) -s can only be obtained as the sum of products of the terms (q i 1 )

γ 1 • • • (q ir ) γr -s with γ 1 i 1 + • • • + γ r i r = l. More precisely, u(p, l, r, i, γ) = (-1) γ 1 +•••+γr γ 1 ! • • • γ r ! (log p) γ 1 +•••+γr .
That this formal computation leads to a convergent Dirichlet series follows from the absolute convergence of the involved Dirichlet series in some half-plane.

Proof of Theorem 3.1. Let ϕ(s) = c 0 s + c 1 + k≥2 c k k -s ∈ C(H) and let (C ϕ j ) ∈ C(H) converging to C ϕ in the operator norm topology. In order to proceed by contradiction, we assume that (char (ϕ j )) does not converge to c 0 . Extracting if necessary, we may assume that char (ϕ j ) = c 0 for all j. We first handle the easiest case c 0 = 0. Since, for all j,

2 -ϕ -2 -ϕ j ≥ lim σ→+∞ |2 -ϕ(σ) -2 -ϕ j (σ) | ≥ 2 -c 1 > 0,
we get that (C ϕ j ) cannot converge to C ϕ . Hence, we assume c 0 ≥ 1. We write ϕ j (s) = d 0 (j)s + d 1 (j) + k≥2 d k (j)k -s . Let p, q be two prime numbers and l ≥ 0. We write

p -ϕ(s) = k≥1 a k k -s and p -ϕ j (s) = k≥1 b k (j)k -s .
Let us first assume that char (ϕ j ) > c 0 for an infinite number of integers j, hence for all j by extraction. Then a p c 0 = p -c 1 whereas b p c 0 (j) = 0. This contradicts that p -ϕ j tends to p -ϕ in H.

Therefore it remains to handle the case char (ϕ j ) < c 0 for all j. Upon another extraction we may and shall assume that d 0 (j) = d 0 ∈ {0, . . . , c 0 -1}. When p = q, for all l ≥ 1, a p d 0 q l = 0 whereas b p d 0 q l (j) = p -d 1 (j) α q l (d(j)). Hence

p = q, l ≥ 1 =⇒ p -d 1 (j) α q l (d(j)) j→+∞ ----→ 0. ( 3.1)
When p = q, setting m = c 0 -d 0 , then a q c 0 = q -c 1 whereas b q c 0 (j) = q -d 1 (j) α q m (d(j)). Thus q -d 1 (j) α q m (d(j))

j→+∞ ----→ q -c 1 .

(3.2) We then fix an increasing sequence of prime numbers (q l ) 1≤l≤m+1 such that, for all l = 1, . . . , m, q l+1 > (q l ) l+1 . We show by a finite induction that, for all l = 1, . . . , m, q

-d 1 (j) l d q l m+1 (j) j→+∞ ----→ 0. ( 3.3)
For l = 1, this is (3.1) applied with p = q 1 , q = q m+1 and we use Lemma 3.3 to compute the value of α q m+1 (d(j)). Let us assume that (3.3) has been obtained for l -1 and let us prove it for l. We apply again (3.1) with p = q l and q = q m+1 so that

q -d 1 (j) l -d q l m+1 (j) log(q l )+ (r,i,γ)∈WP l ir<l u(q l , l, r, i, γ)(d q i 1 m+1 (j)) γ 1 • • • (d q ir m+1 (j)) γr j→+∞ ----→ 0.
Now since q l > q γ 1 i 1 • • • q γr ir for all (r, i, γ) ∈ WP l with i r < l, we may write q

-d 1 (j) l |d q i 1 m+1 (j)| γ 1 • • • |d q ir m+1 (j)| γr ≤ |q -d 1 (j) i 1 d q i 1 m+1 (j)| γ 1 • • • |q -d 1 (j) ir d q ir m+1 (j)| γr
and using the induction hypothesis we obtain that

q -d 1 (j) l (d q i 1 m+1 (j)) γ 1 • • • (d q ir m+1 (j)) γr j→+∞ ----→ 0.
This yields (3.3) at rank l.

Let us now conclude. Property (3.2) for q = q m+1 tells us that

q -d 1 (j) m+1 -d q m m+1 (j) log(q m+1 )+ (r,i,γ)∈WPm ir<m u(q m+1 , m, r, i, γ)(d q i 1 m+1 (j)) γ 1 • • • (d q ir m+1 (j)) γr j→+∞ ----→ q -c 1 m+1 .
But since q m+1 ≥ q m , (3.3) implies that q -d 1 (j) m+1 d q m m+1 (j) tends to zero. Furthermore we can repeat the argument of the inductive step to get

q -d 1 (j) m+1 (r,i,γ)∈WPm ir<m u(q m+1 , m, r, i, γ)(d q i 1 m+1 (j)) γ 1 • • • (d q ir m+1 (j)) γr j→+∞ ----→ 0.
This yields the contradiction we are looking for.

Remark 3.4. In fact our proof shows that the map C(H) → N, C ϕ → char(ϕ) is continuous even if we endow C(H) with the weak operator topology.

3.2. Compact composition operators. We now show that the set of compact composition operators with prescribed characteristic is connected. The proof is necessary different from the proof on H 2 (D) since we cannot connect any compact composition operator to a composition operator with constant symbol.

Theorem 3.5. Let c 0 ∈ N 0 . The set of compact composition operators with characteristic equal to c 0 is arcwise connected.

Proof. We fix ϕ ∈ G with char (ϕ) = c 0 such that C ϕ is compact. We first show that there exists an arc of compact composition operators between C ϕ and C φ for some φ ∈ G with φ(C + ) ⊆ C 1 2 +ε for some ε > 0, char ( φ) = c 0 and, writing φ

= c 0 s + ψ, ψ(C + ) is bounded. Define ϕ σ (•) = ϕ(• + σ).
We claim that σ → C ϕσ is continuous (for the operator norm topology). The choice φ = ϕ 1 will answer the problem: for c 0 = 0, ϕ 1 has restricted range, so that

ϕ 1 (C + ) ⊆ C 1 2 +ε ; for c 0 ≥ 1, we even have ϕ(C + ) ⊆ C 1 . Moreover, if ϕ = c 0 s + ψ,
since the abscissa of uniform convergence of ψ is nonpositive, its abscissa of absolute convergence does not exceed 1/2. Hence, ψ(• + 1) is an absolutely convergent Dirichlet series in C + which implies that ψ(C + ) is bounded. Finally, since for any σ > 0, ϕ σ has restricted range, C ϕσ is compact.

Let T σ be the composition operator induced by the map s → s + σ, σ ≥ 0, so that C ϕσ = T σ • C ϕ . Now observe that for a fixed f ∈ H and a fixed σ 0 ≥ 0, T σ f → T σ 0 f provided σ → σ 0 . Furthermore, {C ϕ (f ) : f ≤ 1} is relatively compact in H and the family (T σ ) σ≥0 is equicontinuous since T σ ≤ 1 for all σ ≥ 0. It follows from a standard compactness argument that the pointwise convergence of T σ f to T σ 0 f when σ tend to σ 0 is uniform on {C ϕ (f ) : f ≤ 1}. Namely for all ε > 0 there exists δ > 0 such that

f ≤ 1 and |σ -σ 0 | < δ =⇒ C ϕσ (f ) -C ϕσ 0 (f ) ≤ ε.
This proves the continuity of σ → C ϕσ .

To finish the proof, we only need to prove that, given ε > 0 and C > 0, for two symbols

ϕ 0 = c 0 s + ψ 0 , ϕ 1 = c 0 s + ψ 1 such that ϕ 0 (C + ), ϕ 1 (C + ) are contained in C 1 2 +ε and |ψ 0 (s)|, |ψ 1 (s)| ≤ C for all s ∈ C + , there exists an arc in C(H) between C ϕ 0 and C ϕ 1 consisting in compact composition operators. Define, for λ ∈ (0, 1), ϕ λ = c 0 s + (1 -λ)ψ 0 + λψ 1 . Again it suffices to show that λ → C ϕ λ is continuous since each ϕ λ has restricted range. Pick f ∈ H with f ≤ 1 and write C ϕ λ (f ) -C ϕ λ (f ) 2 = T ∞ |(f • ϕ λ ) χ (0) -(f • ϕ λ ) χ (0)| 2 dm(χ) = T ∞ |f χ c 0 ((ϕ λ ) χ (0)) -f χ c 0 ((ϕ λ ) χ (0))| 2 dm(χ).
Since (ϕ λ ) χ is a vertical limit of ϕ λ , for all χ ∈ T ∞ , (ϕ λ ) χ (0) and (ϕ λ ) χ (0) belong to C1 2 +ε . Since f χ c 0 ≤ 1, there exists some M > 0, depending only on ε, such that, for all χ ∈ T ∞ , sup

s∈C 1 2 +ε |f χ c 0 (s)| ≤ M. Therefore C ϕ λ (f ) -C ϕ λ (f ) 2 ≤ M 2 T ∞ |(ϕ λ ) χ (0)) -(ϕ λ ) χ (0)| 2 dm(χ) |λ -λ | 2 T ∞ |(ψ 0 ) χ (0)| + |(ψ 1 ) χ (0)| 2 dm(χ) |λ -λ | 2 since for all χ ∈ T ∞ , |(ψ 0 ) χ (0)| ≤ C and |(ψ 1 ) χ (0)| ≤ C.

3.3.

Essential norm of the difference and connected components. The aim of this subsection is to give a necessary condition for two composition operators to be in the same connected component. Inspired by [START_REF] Maccluer | Components in the space of composition operators[END_REF], we will deduce it from a minoration of the essential norm of the difference of two composition operators. Usually, this can been done using reproducing kernels. As pointed out in Section 2.3, reproducing kernels are less efficient for composition operators on H and they have been replaced by partial reproducing kernels to study the compactness of composition operators with polynomial symbols. However, Proposition 5 of [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF] is not sufficient in our context, even if we intend to prove that two composition operators with polynomial symbols are not in the same component: an arc between these two operators could involve non polynomial symbols. Therefore, we need supplementary material about partial reproducting kernels. We fix Q a finite subset of P, the set of prime numbers. Let N Q be the set of integers which can be factorized using prime numbers in Q. For ϕ(s)

= c 0 s+c 1 + n≥1 c n n -s , we let ϕ Q (s) = c 0 s + c 1 + n∈N Q c n n -s =: c 0 s + ψ Q (s).
We first observe that ψ Q , which is well-defined on C 1/2 , where ψ Q converges absolutely, extends to C + , and that

ψ Q (C + ) ⊆ C + . Indeed, let us write any z ∈ T ∞ as z = (z Q , w) where z Q = (z i 1 , . . . , z i d ), Q = {p i 1 , . . . , p i d }.
We may define F a.e. on T d by

F (z Q ) = T ∞ Bψ(z Q , w) -1 Bψ(z Q , w) + 1 dm(w) which clearly belongs to H ∞ (T d ) with F ∞ ≤ 1. By looking at the Taylor co- efficients, F = (Bψ Q -1)/(Bψ Q + 1) in a neighbourhood of zero. Therefore Bψ Q extends holomorphically to D d with Re (Bψ Q ) ≥ 0. This yields that ψ Q extends to C + with ψ Q (C + ) ⊆ C + .
By the open mapping theorem,

ψ Q (C + ) ⊆ C + excepts if ψ Q is constant. Obviously when c 0 = 0, if ψ Q is constant, then ψ Q = c 1 and Re (c 1 ) > 1/2. Hence, in all cases, for all Q ⊆ P, ϕ Q ∈ G.
Let us also introduce a variant of partial reproducing kernels. We set, for w ∈ C + ,

K Q,w (s) = p∈Q 1 1 -p -( w+s)
which is a well-defined function on C + , belonging to H, with

K Q,w 2 = p∈Q 1 1 -p -2Re w .
Moreover, it is easy to check for any sequence (s

k ) ⊆ C + with Re (s k ) → 0, K Q,s k / K Q,s k converges weakly to zero.
The following lemma links C ϕ , ϕ Q and K Q,w . Its proof follows exactly that of Proposition 5 in [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF].

Lemma 3.6. Let ϕ ∈ G and let Q ⊆ P finite. Then, for all w ∈ C + ,

C * ϕ (K Q,w ) = K Q,ϕ Q (w) provided c 0 ≥ 1 K ϕ Q (w) provided c 0 = 0.
Let us finally recall the Julia-Caratheodory theorem on the right half-plane C + .

Definition 3.7. Let ϕ : C + → C + be analytic and let α ∈ R. We say that ϕ admits a finite angular derivative at iα provided lim inf s→iα Re ϕ(s) Re s < +∞.

Lemma 3.8. Let ϕ : C + → C + be analytic and let α ∈ R. Assume that ϕ admits a finite angular derivative at iα and let

d := lim inf s→iα Re ϕ(s) Re s < +∞.
Then both ϕ and ϕ have respective nontangential limits ϕ(iα) and ϕ (iα), with ϕ(iα) ∈ iR and ϕ (iα) = d. Moreover, d ∈ (0, +∞), d is the nontangential limit of Re ϕ(s)/Re s and (ϕ(s) -ϕ(iα))/(s -iα) converges nontangentially to ϕ (iα).

We are now ready to give a lower bound for the essential norm of the difference of two composition operators (see [START_REF] Maccluer | Components in the space of composition operators[END_REF] for the corresponding result on the disk). Theorem 3.9. Let ϕ 0 , ϕ 1 ∈ G with positive characteristic. Let Q ⊆ P finite, d = card(Q). Assume that there exists α ∈ iR such that ϕ 0,Q admits a finite angular derivative at iα. Then either ϕ 1,Q admits a finite angular derivative at iα

with ϕ 1,Q (iα) = ϕ 0,Q (iα) and ϕ 1,Q (iα) = ϕ 0,Q (iα) or C ϕ 0 -C ϕ 1 e ≥ 1 ϕ 0,Q (iα) d/2 .
Proof. Without loss of generality, we may and shall assume that iα = 0 and ϕ 0,Q (iα) = 0. We assume that either ϕ 1,Q does not admit a finite angular derivative at 0 or that either

ϕ 1,Q (0) = 0 or ϕ 1,Q (0) = ϕ 0,Q (0). For a sequence (s k ) ⊆ C + with Re (s k ) tending to zero, we set k Q,s k = K Q,s k / K Q,s k
and observe that k Q,s k converges weakly to zero. Therefore, by Lemma 3.6,

C ϕ 0 -C ϕ 1 2 e = C * ϕ 0 -C * ϕ 1 2 e ≥ lim sup k→+∞ C * ϕ 0 (k Q,s k ) -C * ϕ 1 (k Q,s k ) 2 ≥ lim sup k→+∞ K Q,ϕ 0,Q (s k ) 2 + K Q,ϕ 1,Q (s k ) 2 -2Re K Q,ϕ 0,Q (s k ) (ϕ 1,Q (s k )) K Q,s k 2 .
If (s k ) converges nontangentially to zero, using the computation of the norm of the partial reproducing kernel and Julia-Caratheodory theorem, we get

K Q,ϕ 0,Q (s k ) 2 K Q,s k 2 ∼ Re s k Re ϕ 0,Q (s k ) d ∼ 1 ϕ 0,Q (0) d .
Hence, to conclude, we just need to find a sequence (s k ) ⊆ C + tending nontangentially to zero and such that

lim k→+∞ K Q,ϕ 0,Q (s k ) (ϕ 1,Q (s k )) K Q,s k 2 = lim k→+∞ p∈Q 1 -p -2Re (s k ) 1 -p -(ϕ 0,Q (s k )+ϕ 1,Q (s k )) (3.4)
is as small as possible. First of all, if lim σ→0 + ϕ 1,Q (σ) = 0, we consider a sequence (σ n ) of positive real numbers tending to zero and such that |ϕ 1,Q (σ n )| ≥ δ for some δ > 0. Then, setting s k = σ k , it is clear that the limit appearing in (3.4) is equal to zero. Thus we may assume that ϕ 1,Q has radial limit 0 at 0. Let M > 0 and set

s k = 1 k + i M k . We observe that p∈Q 1 -p -2Re (s k ) 1 -p -(ϕ 0,Q (s k )+ϕ 1,Q (s k )) ∼ +∞ ϕ 0,Q (s k ) + ϕ 1,Q (s k ) 2Re s k -d ∼ +∞ s k 2Re (s k ) -d × ϕ 0,Q (s k ) s k × s k s k + ϕ 1,Q (s k ) s k -d .
Provided M is large enough, |s k |/Re (s k ) could become as large as we want when k tends to +∞. On the other hand,

ϕ 0,Q (s k ) s k × s k s k k→+∞ ----→ ϕ 0,Q (0) × 1 -iM 1 + iM .
If ϕ 1,Q does not have finite angular derivative at 0, then

ϕ 1,Q (s k )/s k tends to +∞. If ϕ 1,Q has a finite angular derivative at 0 with ϕ 1,Q (0) = ϕ 0,Q (0), then lim k→+∞ ϕ 0,Q (s k ) s k × s k s k + ϕ 1,Q (s k ) s k = ϕ 0,Q (0) × 1 -iM 1 + iM + ϕ 1,Q (0) ≥ 1 2 ϕ 0,Q (0) -ϕ 1,Q (0)
provided M is sufficiently large. Therefore, in both cases, for any ε > 0, we have shown that we may choose M > 0 such that lim k→+∞ p∈Q

1 -p -2Re (s k ) 1 -p -(ϕ 0,Q (s k )+ϕ 1,Q (s k )) ≤ ε.
This yields the result.

Remark 3.10. Theorem 3.9 is only interesting when ϕ 0 and ϕ 1 have unrestricted range. However, even if ϕ 0 is continuous, it could be possible that there does not exist α ∈ R with Re ϕ(iα) = 0. Nevertheless, even in that case, one can apply Theorem 3.9 to some ϕ χ as in the proof of the fothcoming Theorem 4.5.

Corollary 3.11. Let ϕ 0 , ϕ 1 ∈ G with char (ϕ 0 ) = char (ϕ 1 ) > 0. Assume that C ϕ 0 and C ϕ 1 belong to the same component of C(H). Then for all Q ⊆ P finite and all α ∈ R such that ϕ 0,Q admits a finite angular derivative at iα, ϕ 1,Q admits a finite angular derivative at iα with ϕ

0,Q (iα) = ϕ 1,Q (iα) and ϕ 0,Q (iα) = ϕ 1,Q (iα).
Proof. Let Q ⊆ P finite and let α ∈ R be such that ϕ 0,Q admits a finite angular derivative at iα. Assume that C ϕ 1 belongs to the connected component

C 0 of C ϕ 0 in C(H). For C τ ∈ C 0 , let U τ be the ball (in C(H)) of center C τ and radius 1/2ϕ 0,Q (iα) d/2 , with d = card(Q). There exists a simple chain U τ 1 , . . . , U τn extract- ed from (U τ ) Cτ ∈C 0 with C ϕ 0 ∈ U τ 1 , C ϕ 1 ∈ U τn and U τ i ∩ U τ j = ∅ for i = j if and only if |i-j| ≤ 1. Let C τ i ∈ U τ i ∩U τ i+1 .
Then setting φ0 = ϕ 0 , φi = τ i for i = 1, . . . , n-1, and φn = ϕ 1 , we have

C φi -C ϕ i-1 < 1 ϕ 0,Q (iα) d/2
for all i = 1, . . . , n. By an immediate induction and Theorem 3.9, we get that for all i = 0, . . . , n, φi admits a finite angular derivative at iα with φi (iα) = ϕ 0,Q (iα) and φi (iα) = ϕ 0,Q (iα). This is exactly the desired result for i = n.

Example 3.12. Let ϕ 0 (s) = s + (1 -2 -s ) and ϕ 1 (s) = s + (1 -3 -s ). Then C ϕ 0 and C ϕ 1 are not in the same component of C(H).
We just need to apply Corollary 3.11 with

Q = {2}. Example 3.13. Let ϕ 0 (s) = s. Then C ϕ 0 is isolated in C(H).
Indeed, if C ϕ 1 belongs to the same component of C ϕ 0 , then for all Q ⊆ P, for all α ∈ R, ϕ 1,Q admits an angular derivative at iα with ϕ

1,Q (iα) = ϕ 0,Q (iα) = ϕ 0 (iα) = iα. Hence, ϕ 1,Q (s) = s which in turn implies ϕ 1 (s) = s.
The use of the angular derivative is not as efficient when c 0 = 0. The reason is simple: it is possible that ϕ Q admits a finite angular derivative at iα and that C ϕ is compact. This is for instance the case of

ϕ(s) = 1 2 + (1 -2 -s ) + (1 -3 -s ).
Nevertheless, we still get a result when card(Q) = 1. Observe that at a boundary point iα, we now have ϕ Q (iα) ∈ 1 2 +iR and that we are now interested in (Re

ϕ Q (s)- 1/2)/Re s. Theorem 3.14. Let ϕ 0 , ϕ 1 ∈ G with zero characteristic. Let Q ⊆ P with card(Q) = 1.
Assume that there exists α ∈ iR such that ϕ 0,Q admits a finite angular derivative at iα. Then either ϕ 1,Q admits a finite angular derivative at iα with ϕ

1,Q (iα) = ϕ 0,Q (iα) and ϕ 1,Q (iα) = ϕ 0,Q (iα) or C ϕ 0 -C ϕ 1 e ≥ C Q ϕ 0,Q (iα) 1/2 .
Proof. We only point out the differences with the proof of Theorem 3.9. We have to replace everywhere K Q,ϕ 0,Q (s k ) by K ϕ 0,Q (s k ) . One now has

K w 2 = ζ(2Re (w))
and

K ϕ 0,Q (s k ) (ϕ 1,Q (s k )) = ζ ϕ 0,Q (s k ) + ϕ 1,Q (s k ) .
The estimate of this last term follows that done in Theorem 3.9, except that we have to use

ζ(s) ∼ 1/(s -1) if s → 1 instead of 1 -p -s ∼ -(log p)s if s → 0.
Corollary 3.15. Let ϕ 0 , ϕ 1 ∈ G with char (ϕ 0 ) = char (ϕ 1 ) = 0. Assume that C ϕ 0 and C ϕ 1 belong to the same component of C(H). Then for all Q ⊆ P with card(Q) = 1 and all α ∈ R such that ϕ 0,Q admits a finite angular derivative at iα, ϕ 1,Q admits a finite angular derivative at iα with ϕ 0,Q (iα

) = ϕ 1,Q (iα) and ϕ 0,Q (iα) = ϕ 1,Q (iα). Example 3.16. Let ϕ 0 (s) = 1 2 + (1 -2 -s ) and ϕ 1 (s) = 1 2 + (1 -3 -s
ϕ 0 (s) = s + (2 log 2 + log 3) -log 3 • 2 -s -2 log 2 • 3 -s ϕ 1 (s) = s + (log 2 + 2 log 3) -2 log 3 • 2 -s -log 2 • 3 -s ?
A plausible conjecture would be that Corollary 3.11 could be extended to the angular partial derivatives of the Bohr lifts Bψ 0,Q and Bψ 1,Q . Nevertheless, the curve σ → (2 -σ , . . . , p -σ d ) is not a special curve in D d , which prevents the use of the Julia-Caratheodory theorem of the polydisc (see [START_REF] Abate | The Julia-Wolff-Carathéodory theorem in polydisks[END_REF]).

However, inspecting the partial derivatives of the Bohr lifts will be fruitful for studying compact linear combinations of composition operators with regular symbols, as indicated by the results of the next section.

Compact linear combinations of polynomial symbols

4.1. Lower bound for the esssential norm from second order data. We start with a result saying that if a linear combination

λ 1 C ϕ 1 + • • • + λ N C ϕ N is compact,
then the coefficients λ j should satisfy some properties depending on the behaviour of the Bψ j at their boundary points. We need first to introduce some terminology inspired by [START_REF] Kriete | Linear relations in the Calkin algebra for composition operators[END_REF].

(ξ) = G(ξ) and ∂ α F (ξ) = ∂ α G(ξ) for all α ∈ N d with |α| ≤ 2. We also denote by Γ(F ) = {z ∈ T d : Re F (z) = 0}.
We will need a couple of lemmas. Lemma 4.2. Let d ≥ 1 and let Q be a non-zero quadratic polynomial in d variables (with complex coefficients). Then

Z(Q) = {x ∈ R d : Q(x) = 0} has Lebesgue measure zero.
Proof. The proof is done by induction on d. For d = 1, the result is of course trivial. Assume that the result has been done until d and let us consider Q a quadratic polynomial in d + 1 variables. Fix x d+1 ∈ R and let Q x d+1 (x 1 , . . . , x d ) = Q(x 1 , . . . , x d+1 ). Then either Q x d+1 is identically zero or its zero set has Lebesgue measure zero. Writing

Q(x) = a + d+1 l=1 b l x l + 1≤l≤m≤d+1 c l,m x l x m , the former case implies that a + b d+1 x d+1 + c d+1,d+1 x 2 d+1 = 0 ∀l = 1, . . . , d, b l + c l,d+1 x d+1 = 0.
Since Q itself is assumed to be non-zero, we can suppose, permuting the variables if necessary, that either a = 0 or b d+1 = 0 or c l,d+1 = 0 for some l = 1, . . . , d + 1. Therefore there are at most two values of x d+1 such that Q x d+1 is identically zero. Hence, for almost all

x d+1 ∈ R, R d 1 Z(Q) (x 1 , . . . , x d+1 )dx 1 . . . dx d = 0.
The lemma follows from an application of Fubini's theorem.

Our second lemma is a result about functions which vanish along (0, +∞). Lemma 4.3. Let C ⊆ N be finite, let α > 0, let (λ j ) j∈C be complex numbers, let (µ j ) j∈C be pairwise distinct complex numbers and let d ≥ 1. Assume that, for all M > 0, j,j ∈C λ j λ j (M α + µ j + µ j ) d = 0.

Then λ j = 0 for all j ∈ C.

Proof. Let us consider

F (z) = j,j ∈C λ j λ j (α + z(µ j + µ j )) d .
Then F is an analytic function defined in a neighbourhood of the origin and the assumption tells us that the origin is an accumulation point of the zeros of F . The result now follows from [START_REF] Choe | Compact linear combinations of composition operators induced by linear fractional maps[END_REF]Corollary 3.4].

Theorem 4.4. Let ϕ j (s) = c 0,j s + ψ j be a finite sequence of symbols in G with ψ j involving only the first d prime numbers, j = 1, . . . , N . Assume that each Bψ j extends to a C 2 -function on D d and that char (ϕ 1 ) > 0. Let (λ j ) j=1,...,N be a finite sequence of complex numbers such that N j=1 λ j C ϕ j is compact. Let z ∈ Γ(Bψ 1 ) and let J be the set of j ∈ {1, . . . , N } such that Bψ j and Bψ 1 have the same second order boundary data at z. Then j∈J λ j = 0.

Proof. Without loss of generality, we may and shall assume that z = e := (1, . . . , 1) and that Bψ 1 (e) = 0. Let us write for j = 1, . . . , N ,

Bψ j (z) = A (j) + d l=1 B (j) l (z l -1) + 1≤l≤m≤d C (j) l,m (z l -1)(z m -1) + o( z -e 2 ).
It is a standard fact (see [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF]Lemma 7] that each B (j) l ∈ (-∞, 0]. We set c 0 = c 0,1 and introduce the following subsets of {1, . . . , N }: I = {j : c 0,j = c 0 and ϕ j (0) = 0} = j : c 0,j = c 0 and A (j) = 0 J 1 = {j : c 0,j = c 0 and ϕ j (0) = 0} J 2 = {j : c 0,j = c 0 and c 0,j > 0} =: {j 1 , . . . , j p } J 3 = {j : c 0,j = 0} .

For j ∈ {1, . . . , N }, let us introduce

L j (x 1 , . . . , x d ) = d l=1 B (j) l log(p l )x l Q j (x 1 , . . . , x d ) = 1 2 d l=1 B (j) l (log p l ) 2 x 2 l + 1≤l≤m≤d C (j) l,m log(p l ) log(p m )x l x m .
We define an equivalence relation on I by j ∼ j ⇐⇒ L j = L j . In other words, j ∼ j if and only if Bψ j and Bψ j have the same first order boundary data at e. Denote by C 1 , . . . , C q the equivalence classes for this relation with 1 ∈ C 1 . We define another equivalence relation on C 1 by jRj ⇐⇒ Q j = Q j and we denote by C 1,1 , . . . , C 1,q the equivalence classes for this relation, with 1 ∈ C 1,1 . In other words, jRj if and only if Bψ j and Bψ j have the same second order boundary data. In particular, J is nothing else than C 1,1 . For i = i ∈ {1, . . . , q}, let

E i,i = β ∈ R d : ∀j ∈ C i , ∀j ∈ C i , L j (β) = L j (β)
(we can replace ∀ by ∃ in the previous definition). Similarly for i = i ∈ {1, . . . , q }, let

F i,i = β ∈ R d : ∀j ∈ C 1,i , ∀j ∈ C 1,i , Q j (β) = Q j (β) .
By Lemma 4.2, there exists

β ∈ i =i E i,i ∩ i =i F i,i . Let finally, for 1 ≤ l ≤ d, k ≥ 1 and M > 0, z l (k) = p -M k +i β l √ k l so that z l (k) = 1 + i log(p l )β l √ k - 2M log(p l ) + log 2 (p l )β 2 l 2k + o 1 k .
Therefore,

Bψ j (z(k)) = A (j) + i √ k d l=1 B (j) l log(p l )β l - 1 k M d l=1 B (j) l log(p l ) + 1 2 d l=1 B (j) l log 2 (p l )β 2 l + 1≤l≤m≤d C (j) l,m log(p l ) log(p m )β l β m + o 1 k = A (j) + i √ k L j (β 1 , . . . , β d ) - 1 k (M L j (e) + Q j (β 1 , . . . , β d )) + o 1 k .
We now choose a sequence of real numbers (t k ) such that

(1) for 1 ≤ l ≤ d, p -it k l is very close to p iβ l √ k
l , so close that, setting

w k = M k + it k , Bψ j (p -w k 1 , . . . , p -w k d ) = Bψ j (z(k)) + o 1 k for all j = 1, . . . , N .
(2) For all j ∈ J 2 , writing j = j r with 1 ≤ r ≤ p, p

-it k (c 0,jr -c 0 ) d+r p -A (jr )
d+r tends to some ω(j r ) = 1. These two conditions may be simultaneously satisfied because of Kronecker's theorem and since c 0,j = c 0 for all j ∈ J 2 . We set d = d + p and use that

N j=1 λ j C ϕ j e ≥ lim sup k→+∞ N j=1 λ j C * ϕ j (k d ,w k ) 2 ≥ lim sup k→+∞ 1≤j,j ≤N λ j λ j C * ϕ j (k d ,w k ), C * ϕ j (k d ,w k ) .
We claim that if j ∈ I and j / ∈ I, then

C * ϕ j (k d ,w k ), C * ϕ j (k d ,w k ) k→+∞ ----→ 0.
First assume that j ∈ J 3 . Then

C * ϕ j (k d ,w k ), C * ϕ j (k d ,w k ) = K ϕ j (w k ) , K d ,ϕ j (w k ) K d ,w k 2 = K d ,ϕ j (w k ) , K d ,ϕ j (w k ) K d ,w k 2
and this goes to zero since

K d ,ϕ j (w k ) is bounded because Re ϕ j (w k ) ≥ 1/2. If j ∈ I and j / ∈ J 3 , then C * ϕ j (k d ,w k ), C * ϕ j (k d ,w k ) = d l=1 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l . Pick j ∈ J 1 . Then ϕ j (w k ) + ϕ j (w k ) → A (j ) = 0 so that d l=1 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l k→+∞ ----→ 0. (4.1)
Pick finally j ∈ J 2 and write j = j r for 1 ≤ r ≤ p. Then

ϕ j (w k ) + ϕ j (w k ) = (c 0 -c 0,jr )it k + A (jr) + o(1) so that p -(ϕ j (w k )+ϕ j (w k ) d+r k→+∞ ----→ ω(j ) = 1. Therefore, 1 -p -2Re (w k ) d+r 1 -p -(ϕ j (w k )+ϕ j (w k )) d+r k→+∞ ----→ 0.
On the other hand, since Re ϕ j (w k ), Re ϕ j (w k ) ≥ Re (w k ), we also have

d 1≤l≤d , l =d+r 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l ≤ 1.
Hence, (4.1) remains true for these values of j and j . This implies that 0

≥ N j=1 λ j C ϕ j e ≥ lim sup k→+∞ j,j ∈I λ j λ j d l=1 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l + j,j / ∈I λ j λ j d l=1 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l .
Since each term in the previous sum is nonnegative (it is equal to

j∈E λ j C * ϕ j (k d,w k ) 2
for either E = I or E = {1, . . . , N }\I), this yields lim sup k→+∞ j,j ∈I

λ j λ j d l=1 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l = 0.
Consider now j, j ∈ I, and observe that

ϕ j (w k ) + ϕ j (w k ) = 2c 0 M k + Bψ j (z(k)) + Bψ j (z(k)) + o 1 k = i(L j (β) -L j (β)) √ k + 2c 0 M k - M (L j (e) + L j (e)) k - Q j (β) + Q j (β) k + o 1 k . If j ∈ C i and j ∈ C i with i = i , so that L j (β) = L j (β), then ϕ j (w k ) + ϕ j (w k ) ∼ k→+∞ i(L j (β) -L j (β))
√ k and this implies easily that, for all l ≥ 1,

1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l k→+∞ ----→ 0. If j = j ∈ C 1 , one now gets that 1 -p -2Re (w k ) l 1 -p -(ϕ j (w k )+ϕ j (w k )) l k→+∞ ----→ M 2M (c 0 -L 1 (e)) -(Q j (β) + Q j (β)) .
Arguing as above, we finally find that, for all M > 0,

j,j ∈C 1 λ j λ j M d (2M (c 0 -L 1 (e)) -(Q j (β) + Q j (β)) d = 0. (4.2)
Let for k = 1, . . . , q , ν k = j∈C 1,k λ j and let µ k = Q j (β) for some (all) j ∈ C 1,k , 1 ≤ k ≤ q . Then (4.2) may be rewritten

q k,k =1 ν k ν k M d 2M (c 0 -L 1 (e)) -(µ k + µ k ) d = 0.
Lemma 4.3 tells us that ν k = 0 for all k = 1, . . . , q . The case k = 1 is exactly the required statement.

4.2. Linear combinations of polynomial symbols with degree at most 2.

We are now ready for the proof of our extension of the main result of [START_REF] Bayart | Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series[END_REF].

Theorem 4.5. Let (ϕ j ) j=1,...,N ⊆ G be a finite sequence of distinct symbols. Assume that each ϕ j is a polynomial symbol with degree at most 2. Let also (λ j ) j=1,...,N be a finite sequence of non-zero complex numbers. Then N j=1 λ j C ϕ j is compact on H if and only if each C ϕ j is compact.

Proof. We proceed by induction on N , the case N = 1 being trivial. Assume that the result has been obtained for N -1 symbols and let us prove it for N symbols. By contradiction, assume that no C ϕ j is compact. If one symbol, say ϕ 1 , has a positive characteristic, then we can conclude immediately using Theorem 4.4. Indeed, since ϕ 1 has unrestricted range (it is assumed to be noncompact), Γ(Bψ 1 ) = ∅ (the polycircle T d is compact). Now, for j = 1, Bψ 1 and Bψ j cannot have the same second order boundary data at some point, since ϕ 1 and ϕ j are distinct polynomials of degree at most 2. Thus, we get λ 1 = 0, a contradiction.

Therefore, we are reduced to the case where all symbols ϕ j have zero characteristic. Let us fix j ∈ {1, . . . , N }. In the terminology of [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF], the complex dimension of ϕ j cannot be greater than or equal to 2: since the degree of ϕ j is at most 2, Theorem 3 of [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF] would imply that C ϕ j is compact. Thus we have two possibilities for ϕ j : either ϕ j (s) = c 1,j + d j q -s 1,j q -s 2,j with q 1,j = q 2,j ∈ P, d j = 0 (4.3) which corresponds to the case where the complex dimension and the degree of ϕ 2 are equal to 2, or ϕ j (s) = c 1,j + e j q -s j + f j q -2s j with q j ∈ P (4.4) which means that ϕ j has complex dimension 1 and degree 1 or 2 (provided f j = 0 or not). If all symbols may be written as in (4.3), then we face to linear symbols and this case will be handled in the forthcoming Theorem 4.8. Therefore we assume that at least one symbol, say ϕ 1 , has a form like in (4.4). Let χ ∈ T ∞ . Since

(f • ϕ j ) χ = f • (ϕ j ) χ , the following diagram commutes H j λ j Cϕ j ------→ H U     V H j λ j C (ϕ j )χ
-------→ H where U and V are the unitary maps of H defined by U (f ) = f and V (f ) = f χ . Therefore, the compactness of λ j C ϕ j is equivalent to that of j λ j C (ϕ j )χ which implies that we may assume that Re (ϕ 1 (0)) = 1/2. By vertical translations, we may even assume that ϕ 1 (0) = 1/2. Let I = j : ϕ j may be written as in (4.4) with q j = q 1 , ϕ j (0) = 1 2 , ϕ j (0) = ϕ 1 (0)

J 1 = j : ϕ j may be written as in (4.4) with q j = q 1 , ϕ j (0) = 1 2 , ϕ j (0) = ϕ 1 (0)

J 2 = j : ϕ j may be written as in (4.4) with q j = q 1 , ϕ j (0) = 1 2

J 3 = j : ϕ j may be written as in (4.4) with q j = q 1 J 4 = j : ϕ j may be written as in (4.3) .

Let finally s

k = 1 k + i M √
k for some M > 0 and consider the normalized partial reproducing kernels k s k = K (q 1 ) s k / K (q 1 ) s k ; the sequence (k s k ) goes weakly to zero. By Lemma 3.6, if

j ∈ I ∪ J 1 ∪ J 2 , C * ϕ j (k s k ) = K ϕ j (s k ) K (q 1 ) s k whereas, if j ∈ J 3 ∪ J 4 , C * ϕ j (k s k ) = K c 1,j K (q 1 ) s k k→+∞ ----→ 0.
Using this last result, one gets

N j=1 λ j C ϕ j 2 e ≥ lim sup k→+∞ N j=1 λ j C * ϕ j (k s k ) 2 ≥ lim sup k→+∞ j,j ∈I∪J 1 ∪J 2 λ j λ j K ϕ j (s k ) , K ϕ j (s k ) K (q 1 ) s k 2 ≥ lim sup k→+∞ j,j ∈I∪J 1 ∪J 2 λ j λ j ζ(ϕ j (s k ) + ϕ j (s k )) K (q 1 ) s k 2 .
We claim that, for all j ∈ I and all j ∈ J 1 ∪ J 2 ,

ζ(ϕ j (s k ) + ϕ j (s k )) K (q 1 ) s k 2 k→+∞ ----→ 0. (4.5)
Let us observe that, for all j ∈ {1, . . . , N },

ϕ j (s k ) = ϕ j (0) + iM ϕ j (0) √ k + 2ϕ j (0) -M 2 ϕ j (0) 2k + o 1 k so that if j ∈ I and j ∈ I ∪ J 1 ∪ J 2 , ϕ j (s k ) + ϕ j (s k ) = 1 2 + ϕ j (0) + i ϕ j (0) -ϕ j (0) √ k M + 2ϕ j (0) + 2ϕ j (0) -M 2 (ϕ j (0) + ϕ j (0)) 2k + o 1 k .
Hence, (4.5) follows now easily from the properties of ζ and K (q 1 ) s k 2 ∼ Ak. On the other hand, when j, j ∈ I, one gets

ζ(ϕ j (s k ) + ϕ j (s k )) K (q 1 ) s k 2 k→+∞ ----→ 2A 4ϕ 1 (0) -M 2 (ϕ j (0) + ϕ j (0)) .
Therefore, arguing as in the proof of Theorem 4.4, we get that for all M > 0, j,j ∈I

λ j λ j 4ϕ 1 (0) -M 2 (ϕ j (0) + ϕ j (0)) = 0.
Now, for j = j ∈ I, one has ϕ j (0) = ϕ j (0). Indeed, if j, j ∈ I with ϕ j (0) = ϕ j (0), then ϕ j (0) = -(log q 1 )e j -2(log q 1 )f j = -(log q 1 )e j -2(log q 1 )f j = ϕ j (0) ϕ j (0) = (log q 1 ) 2 e j + 4(log q 1 ) 2 f j = (log q 1 ) 2 e j + 4(log q 1 ) 2 f j = ϕ j (0).

which implies e j = e j and f j = f j , thus ϕ j = ϕ j and j = j . Finally, Lemma 4.3 gives λ j = 0 for j ∈ I, a contradiction.

4.3.

Linear combinations of linear symbols. Theorem 4.5 does not encompass completely the main result of [START_REF] Bayart | Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series[END_REF]. Indeed, for instance, ϕ(s) = s + 1 -30 -s is a linear symbol which is not a polynomial symbol of degree at most 2. Nevertheless, we are also able to handle these symbols, thanks to the two forthcoming theorems.

Theorem 4.6. Let (ϕ j ) j=1,...,N ⊆ G be a finite sequence of distinct linear symbols such that at least one of them has a positive characteristic. Let (λ j ) j=1,...,N be a finite sequence of non-zero complex numbers. Then N j=1 λ j C ϕ j is compact on H if and only if each C ϕ j is compact.

Proof. Arguing as at the beginning of the proof of Theorem 4.5, we can assume that ϕ 1 has a positive characteristic, and N j=1 λ j C ϕ j is compact whereas no C ϕ j is compact. Then it will be shown that λ 1 = 0, which leads to a contradiction with the hypotheses. We can also assume that ϕ 1 (0) = 0. We write each ϕ j as

ϕ j (s) = c 0,j s + c 1,j + d (j) l=1 c q (j) l (q (j) l ) -s = c 0,j s + ψ j (s)
where the integers (q (j) l ) j=1,...,d (j) are multiplicatively independent and the complex numbers c q (j) l are non-zero. Since ϕ j is supposed to have unrestricted range and the (q (j) l ) are multiplicatively independent, we know that Re (c 1,j ) = d (j) l=1 |c q (j) l | if c 0,j = 0, and Re (c 1,j ) = 1 2 + d (j) l=1 |c q (j) l | otherwise. To simplify the notation, we set c 0 = c 0,1 , d = d (1) , q l = q

(1) l , c q l = c q (1) l and Q = {q 1 , . . . , q d }. We set

Q N = {q ∈ N : ∃α ∈ Q d , q = q α 1 1 • • • q α d d }
and observe that if q / ∈ Q N , then the (d + 1) integers q 1 , . . . , q d , q are multiplicatively independent.

Inspired by the proof of Theorem 4.5, we set

I = j : c 0,j = c 0 , ϕ j (0) = 0, q (j) l ∈ Q N , l = 1, . . . , d (j) J 1 = j : c 0,j = c 0 , ϕ j (0) = 0, q (j) l ∈ Q N , l = 1, . . . , d (j) J 2 = j : c 0,j = c 0 , ∃l ∈ {1, . . . , d (j) }, q (j) l / ∈ Q N J 3 = {j : c 0,j = c 0 and c 0,j > 0} =: {j 1 , . . . , j p } J 4 = {j : c 0,j = 0} .
We start by working with indices in I. Therefore, let j ∈ I. The map ϕ j may be written

ϕ j (s) = c 0 s + α∈Γ j c (j) α (q α ) -s -1 with Γ j ⊆ Q d \{(0, . . . , 0)} and c (j)
α < 0 for all α ∈ Γ j . In particular

B Q ψ j (z) = α∈Γ j c (j) α d l=1 z α l l -1 .
For j ∈ I, let us introduce

L j (x 1 , . . . , x d ) = d l=1 α∈Γ j c (j) α α l log(p l )x l Q j (x 1 , . . . , x d ) = 1 2 1≤l,m≤d α∈Γ j c (j) α α l α m log(p l ) log(p m )x l x m .
Let also C = {j ∈ I :

L j = L 1 } and pick j ∈ I, j = 1. If Q j = Q 1 , then for all 1 ≤ l = m ≤ d, α∈Γ j c (j) α α l α m = 0 (4.6)
for all 1 ≤ l ≤ d,

α∈Γ j c (j) α α 2 l = c q l . (4.7)
Since each c (j) α is negative, (4.6) tells us that for any α ∈ Γ j , at most one of its component is non-zero. Moreover, since the integers q α , α ∈ Γ j , are multiplicatively independent, if l ∈ {1, . . . , d}, at most one α ∈ Γ j can have its non-zero component equal to its l-th component. Therefore, the sum appearing in (4.7) reduces to at most one element. Since c q l = 0, the sum cannot be empty and we have shown that card(Γ j ) = d, that Γ j may be written {α(1), . . . , α(d)} with α(l) = (0, . . . , 0, α l (l), 0, . . . , 0), the non-zero term α l (l) appearing at the l-th position. Now, if we compare (4.7) with α l (l)d (j) α(l) = α∈Γ j α l c (j) α = c q l (which comes from L j = L 1 ), we get that each α l (l) has to be equal to 1. Finally, we have shown that Γ j = {(1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, . . . , 0, 1)} which in turn yields ϕ j = ϕ 1 , a contradiction.

From this, we deduce using an argument similar to that of the proof of Theorem 4.5 the existence of β ∈ R d such that, for all j ∈ I, j = 1, either

L j (β) = L 1 (β) or Q j (β) = Q 1 (β). We now set, for 1 ≤ l ≤ d, k ≥ 1 and M > 0, z l (k) = p -M k + iβ l √ k l so that z l (k) α l = 1 + i α l log(p l )β l √ k - M α l log(p l ) + 1 2 α 2 l log 2 (p l )β 2 l k + o 1 k .
This yields for any j ∈ I,

B Q ψ j (z(k)) = i √ k α∈Γ j c (j) α d l=1 α l log(p l )β l - 1 k α∈Γ j c (j) α d l=1 M α l log(p l ) + 1 2 d l=1 α 2 l log 2 (p l )β 2 l + 1≤l<m≤d c (j) α α l α m log(p l ) log(p m )β l β m + o 1 k = iL j (β) √ k - L j (e) + Q j (β) k + o 1 k .
Suppose now that j ∈ J 1 which means that B Q ψ j (e) = 0. Hence, B Q ψ j (z(k)) tends to some a j = 0. To handle J 2 , we first introduce integers q d+1 , . . . , q e such that q 1 , . . . , q d , q d+1 , . . . , q e is a maximal sequence of multiplicatively independent integers extracted from {q

(j) l : 1 ≤ j ≤ N, 1 ≤ l ≤ d (j) }. For j ∈ J 2 , each q (j) l may be written q α 1 1 • • • q α d d q α d+1 d+1 • • • q αe e with α i ∈ Q.
Furthermore, there exists some l(j) such that the family (α d+1 , . . . , α e ) is not identically zero (otherwise j would belong to I ∪ J 1 ). Denote this family by (α d+1 (j), . . . , α e (j)). Then

E j := (θ d+1 , . . . , θ e ) ∈ R e-d : e i(θ d+1 α d+1 (j)+•••+θeαe(j) c q (l) j = |c q (j) l |
has Lebesgue measure zero. Therefore we can pick (θ d+1 , . . . , θ e ) ∈ R e-d \ j∈J 2 E j .

Denoting Q = {q d+1 , . . . , q e }, since Re (c 1,j ) = d (j) l=1 |c q (j) l |, this implies that B Q∪Q ψ j (1, . . . , 1, e iθ d+1 , . . . , e iθe ) =: a j = 0.

To proceed with the conclusion, we finally fix some positive integers q e+1 , . . . , q e+p such that q 1 , . . . , q e+p are multiplicatively independent (recall that p = card(J 3 )).

We then choose, by Kronecker's theorem, a sequence of real numbers (t k ) such that

(a) for 1 ≤ l ≤ d, q -it k l is very close to q - iβ l √ k l , so close that, setting w k = -M k + it k , B Q ψ j (q -w k 1 , . . . , q -w k d ) = B Q ψ j (z(k)) + o 1 k for all j ∈ I ∪ J 1 . (b) for d + 1 ≤ l ≤ e, q -it k l
tends to e iθ l , so that

B Q∪Q ψ j (q -w k 1 , . . . , q -w k e ) = a j + o 1 k
for all j ∈ J 2 . (c) for all j ∈ J 3 , writing j = j r with 1 ≤ r ≤ p, q -it k (c 0,jr -c 0 ) e+r q -B Q∪Q ψ j (1,...,1,e iθ d+1 ,...,e iθe ) e+r tends to some ω(j r ) = 1.

Let d be such that p + (q l ) ≤ p d for all l = 1, . . . , e + r and p + (q (j) l ) ≤ p d for all j ∈ J 4 and all 1 ≤ l ≤ d (j) . We first observe that, for j ∈ C and j / ∈ C,

C * ϕ j (k d ,w k ), C * ϕ j (k d ,w k ) k→+∞ ----→ 0.
When j / ∈ J 4 , this follows from the choice of the sequence t k , and the fact that for l ∈ {1, . . . , N }, l / ∈ J 4 ,

ϕ l (w k ) = c 0,l w k + B Q∪Q ψ l (q -w k 1 , . . . , q -w k e ).
When j ∈ J 4 , this has already been observed during the proof of Theorem 4.4. We now argue exactly as in the proof of Theorem 4.4. to conclude that, for all M > 0, j,j ∈C

λ j λ j 2M (c 0 -L 1 (e)) -(Q j (β) + Q j (β)) d = 0. (4.8)
Finally we consider the following equivalence relation on C: jRj ⇐⇒ Q j (β) = Q j (β). Let C 1 , . . . , C κ be the equivalent classes for this relation, with

C 1 = {1}, let ν k = j∈C k λ j and let µ k = Q j (β) for some (all) j ∈ C k , 1 ≤ k ≤ κ. Then (4.8) may be rewritten κ k,k =1 ν k ν k 2M (c 0 -L 1 (e)) -(µ k + µ k ) d = 0.
Lemma 4.3 tells us that, for all k = 1, . . . , κ, ν k = 0. In particular,

λ 1 = ν 1 = 0, a contradiction!
To tackle the case where all symbols have zero characteristic, we need one elementary lemma: Lemma 4.7. Let Q be a finite set of positive integers. Assume that for any n, m ∈ Q, there exist α, β ∈ N such that n α = m β . Then there exists q ∈ N such that, for any n ∈ Q, there exists α ∈ N with n = q α . Proof. We proceed by induction on the cardinal N of Q, the result being trivial for N = 1 and N = 2. Let us assume that it has been proved until N and let Q = {q 1 , . . . , q N +1 } be a set with N + 1 elements. We apply the inductive step to find some q working with Q = {q 1 , . . . , q N } and then we apply the result for N = 2 to {q , q N +1 } to conclude. Theorem 4.8. Let (ϕ j ) j=1,...,N ⊆ G be a finite sequence of distinct linear symbols with zero characteristic. Let (λ j ) j=1,...,N be a finite sequence of non-zero complex numbers. Then N j=1 λ j C ϕ j is compact on H if and only if each C ϕ j is compact. Proof. Arguing as above, we start with N linear symbols ϕ 1 , . . . , ϕ N having zero characteristic and with N non-zero complex numbers λ 1 , . . . , λ N such that N j=1 λ j C ϕ j is compact whereas no C ϕ j is compact. In particular, one may write ϕ j (s) = c j + d j q -s j where |d j | = Re (c j ) -1 2 . Without loss of generality, we shall assume that ϕ 1 (0) = 1 2 . As before, we split {1, . . . , N } into disjoint subsets: I = j : ϕ j (0) = ϕ 1 (0) and q α j = q β 1 for some α, β ∈ N J 1 = j : ϕ j (0) = ϕ 1 (0) and q α j = q β 1 for some α, β ∈ N J 2 = j : q α j = q β 1 for all α, β ∈ N .

Let q be given by Lemma 4.7 with Q = {q j : j ∈ I ∪ J 1 }, for j ∈ I, let α j ∈ N be such that q j = q α j and, for M > 0, consider

w k = M k + i √ k . When j ∈ I, ϕ j (w k ) = c j + d j q -α j w k -1 + d j = 1 2 - iα j d j log q √ k - 2M α j d j log q + α 2 j d j log 2 q 2k + o 1 k
and let us recall that d j < 0. Moreover, when j ∈ J 2 , applying Lemma 2.1,

C * ϕ j (k (q) w k ) = K c j K (q) w k k→+∞ ----→ 0.
Therefore, one gets

0 ≥ N j=1 λ j C ϕ j 2 e ≥ lim sup k→+∞ N j=1 λ j C * ϕ j (k (q) w k ) 2 ≥ lim sup k→+∞ j∈I∪J 1 λ j C * ϕ j (k (q) w k ) 2 ≥ lim sup k→+∞ j,j ∈I∪J 1 λ j λ j K ϕ j (w k ) (ϕ j (w k )) K (q) w k 2 ≥ lim sup k→+∞ j,j ∈I∪J 1 λ j λ j ζ ϕ j (w k ) + ϕ j (w k ) K (q) w k 2
. Now, if j ∈ I and j ∈ J 1 , ϕ j (w k ) + ϕ j (w k ) tends to some complex number in C 1 which is not equal to 1. Hence,

ζ ϕ j (w k ) + ϕ j (w k ) K (q) w k 2 k→+∞ ----→ 0
so that, arguing as in the proof of Theorem 4.5, lim sup k→+∞ j,j ∈I

λ j λ j ζ ϕ j (w k ) + ϕ j (w k ) K (q) w k 2 = 0. Let finally C = {j ∈ I : α j d j = α 1 d 1 }. Using ζ(s) ∼ s→1 1 s-1 and K (q) w 2 ∼ Re w→0
Cq Re w , we finally get

j,j ∈C λ j λ j 4M α 1 d 1 log q + α 2 j d j log 2 q + α 2 j d j log 2 q = 0.
Since for all j = j ∈ C, it is clear that α 2 j d j = α 2 j d j (otherwise α j = α j and d j = d j ) we conclude by Lemma 4.3 that all λ j , j ∈ C, are equal to zero, a contradiction.

5.

Weigthed composition operators, connected components and compact differences 5.1. Introduction. Our aim in this section is to give sufficient conditions to prove that two composition operators C ϕ 0 and C ϕ 1 belong to the same connected component of C(H), or that their difference is compact. We shall see that this could be deduced from results on weighted composition operators (see [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF] where this idea seems to appear for the first time). We start with ϕ 0 , ϕ 1 ∈ G with the same characteristic c 0 . The simplest way to draw an arc between C ϕ 0 and C ϕ 1 is to consider λ ∈ [0, 1] → C ϕ λ , where ϕ λ = (1 -λ)ϕ 0 + λϕ 1 . To prove that λ → C ϕ λ is continuous, we start as in the proof of Theorem 3.5: let µ be a probability measure on R, f ∈ H with f ≤ 1 and let us write

C ϕ λ (f ) -C ϕ λ (f ) 2 = T ∞ R |f χ c 0 • (ϕ λ ) χ (it) -f χ c 0 • (ϕ λ ) χ (it)| 2 dµ(t)dm(χ).
At this point we do not use the mean value theorem but instead we write

λ λ f χ c 0 ((ϕ r ) χ (it))dr = 1 (ϕ 1 ) χ (it) -(ϕ 0 ) χ (it) (f χ c 0 • (ϕ λ ) χ (it) -f χ c 0 • (ϕ λ ) χ (it))
so that, using Jensen's inequality,

C ϕ λ (f ) -C ϕ λ (f ) 2 ≤ |λ -λ| 2 λ λ T ∞ 1 0 |(ϕ 1 ) χ (it) -(ϕ 0 ) χ (it)| 2 (5.1) ×|f χ c 0 ((ϕ r ) χ (it))| 2 dµ(t)dm(χ)dr.
This inequality leads us to the following lemma.

Lemma 5.1. Let ϕ 0 , ϕ 1 ∈ G with char (ϕ 0 ) = char (ϕ 1 ) and let

ϕ λ = (1-λ)ϕ 0 +λϕ 1 , λ ∈ [0, 1].
(i) Suppose that for all λ ∈ [0, 1], f → (ϕ 0 -ϕ 1 )C ϕ λ (f ) acts boundedly on H with uniformly bounded norms. Then there is a constant A > 0 such that, for all λ, λ ∈ [0, 1],

C ϕ λ -C ϕ λ ≤ A|λ -λ |. (ii) Suppose moreover that for each λ ∈ [0, 1], f → (ϕ 0 -ϕ 1 )C ϕ λ (f ) is a compact operator on H. Then C ϕ 0 -C ϕ 1 is compact.
Proof. (i) is an immediate consequence of (5.1) whereas (ii) follows from an application of Lebesgue's dominated convergence theorem.

Hence we are reduced to study the boundedness and the compactness of a weighted composition and derivation operator. By working with a Bergman space instead of a Hardy space as initial space, we will be reduced to study the same problems for weighted composition operators acting between different spaces. At this point, we will divide the proof into two cases, following the value of c 0 . For c 0 > 0 we will stay on the halfplane (and on the hidden polydisc) and use Nevanlinna counting functions. For c 0 = 0 we will work on the disk and use Carleson measures. 5.2. The case of positive characteristic.

5.2.1.

Boundedness and compactness of weighted composition operators. We shall need the following lemma which explains how to compute the norm of an element of H using its derivative. Lemma 5.2. Let µ be a probability measure on R and f = n a n n -s . Then the following assertions are equivalent:

(i) f ∈ H;

(ii) for almost all χ ∈ T ∞ , f χ extends to C + and

T ∞ R 1 0 |f χ (σ + it)| 2 σdσdµ(t)dm(χ) < +∞.
Moreover, provided f ∈ H, then

f 2 ≈ |a 1 | 2 + T ∞ R 1 0 |f χ (σ + it)| 2 σdσdµ(t)dm(χ). (5.2)
Proof. We just need to prove (5.2) for a Dirichlet polynomial. Now by (2.1) and Fubini's theorem,

T ∞ R 1 0 |f χ (σ + it)| 2 σdσdµ(t)dm(χ) = n≥2 |a n | 2 log 2 (n) 1 0 σn -2σ dσ ≈ n≥2 |a n | 2 .
This leads us to introduce the following Banach space of Dirichlet series: A is the completion of the space of Dirichlet polynomials for the norm

f 2 A := 1 0 f (• + σ) 2 H σdσ = T ∞ R 1 0 |f χ (σ + it)| 2 σdσdµ(t)dm(χ). (5.3)
This space, a particular example of the Bergman spaces introduced in [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF], is a space of Dirichlet series defined on C 1/2 . Moreover, if f ∈ A, for almost all χ ∈ T ∞ , f χ converges on C + and the norm in A can be computed using (5.3).

Lemma 5.2 says that f ∈ H if and only if f ∈ A and that up to the constant term, the norms are comparable. Moreover, it is easy to check that if a sequence (f n ) converges weakly to 0 in H, then (f n ) converges weakly to 0 in A. Therefore we are led to the study of the boundedness and of the compacness of weighted composition operators from A to H. This is the content of the next theorem, of independent interest. The proof will need the following analogue of Lemma 5.2 for A, which has a completely similar proof: for all f ∈ A, for all c 0 ≥ 1,

f 2 A ≈ |a 1 | 2 + T ∞ R +∞ 0 |f χ c 0 (σ + it)| 2 σ 3 dσdµ(t)dm(χ).
Proof of Theorem 5.3. Let f ∈ A. We choose for µ the probability measure dµ = 1 [0,1] dt. We intend to show that uC ϕ (f ) H c 0 ,C f A . By Lemma 5.2, it suffices to show that (uC ϕ (f )) A c 0 ,C f A . Now,

(uC ϕ f ) = u f • ϕ + uϕ f • ϕ.
Therefore we have to study the following integrals:

I 1 = T ∞ 1 0 1 0 |u χ (σ + it)| 2 |f χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) I 2 = T ∞ 1 0 1 0 |u χ (σ + it)| 2 |ϕ χ (σ + it)| 2 |f χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ).
We first handle I 1 and observe that, u χ being a vertical translate of u , for all χ ∈ T ∞ , for all s ∈ C + , |u χ (s)| ≤ C. On the other hand, we appeal to [START_REF] Bailleul | Composition operators on weighted Bergman spaces of Dirichlet series[END_REF] to ensure that

I 1 ≤ C f • ϕ A ≤ C f A .
Let us turn to I 2 . Using again vertical translates, we know that, for all s ∈ C + , |u χ (s)| ≤ CRe ϕ χ (s) . Since we also know that Re (s) ≤ Re ϕ χ (s) /c 0 , we obtain

I 2 c 0 ,C T ∞ 1 0 1 0 |ϕ χ (σ + it)| 2 |f χ c 0 • ϕ χ (σ + it)| 2 Re 3 ϕ χ (σ + it) dσdtdm(χ).
We then do, for a fixed χ ∈ T ∞ , the nonunivalent change of variables w = ϕ χ (s). We denote by N ϕχ,3 the appropriated Nevanlinna counting function, namely

N ϕχ,3 (w) = ϕχ(w)=s Re 3 (s).
Observing that our assumptions ensure that for all χ ∈ T ∞ , ϕ χ (

[0, 1] 2 ) ⊆ [0, c 0 + C] × [-c 0 -C, c 0 + C],
we then get

I 2 c 0 ,C T ∞ c 0 +C -c 0 -C c 0 +C 0 |f χ c 0 (σ + it)| 2 N ϕχ,3 (σ + it)dσdtdm(χ).
I 2 c 0 ,C T ∞ C+c 0 -C-c 0 c 0 +C 0 |f χ c 0 (σ + it)| 2 σ 3 dσdtdm(χ) c 0 ,C f 2 A .
Let us now turn to compactness. Proof. Let (f n ) be a sequence in A which converges weakly to 0 and satisfies f n A ≤ 1. Arguing as in the proof of Theorem 5.3, we have to show that I n,1 and I n,2 go to zero where

I n,1 = T ∞ 1 0 1 0 |u χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ), I n,2 = T ∞ 1 0 1 0 |u χ (σ + it)| 2 |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ).
We first handle I n,1 . Let ε > 0 and θ ∈ (0, 1) be such that, for all χ ∈ T ∞ and for all s ∈ C + with Re ϕ χ (s)

∈ (0, θ), |u χ (s)| ≤ ε. Then, setting Ω χ = {(σ, t) ∈ [0, 1] 2 : Re (ϕ χ (s)) ∈ (0, θ)}, T ∞ Ωχ |u χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) ≤ ε 2 T ∞ 1 0 1 0 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) ≤ ε 2 .
On the other hand, for all χ ∈ T ∞ , for all (σ, t) ∈ [0, 1] 2 \Ω χ , setting s = σ + it, Re (ϕ χ (s)) ≥ θ and |Im (ϕ χ (s))| ≤ C + c 0 . Therefore, there exists C θ > 0 such that, for all χ ∈ T ∞ and all such (σ, t),

|f n,χ c 0 • ϕ χ (σ + it)| 2 ≤ C θ f n,χ c 0 (• + θ/2) 2 H 2 i (C + )
, where H 2 i (C + ) is the invariant Hardy space on the right halfplane, namely the set of analytic functions g :

C + → C + satisfying g 2 H 2 i (C + ) := 1 π +∞ -∞ |g(it)| 2 dt 1 + t 2 < +∞.
This yields

T ∞ [0,1] 2 \Ωχ |u χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) θ T ∞ 1 0 1 0 R f n,χ c 0 iu + θ 2 2 du 1 + u 2 σdσdtdm(χ) θ k≥1 |a n,k | 2 k -θ ,
where we have written each f n as f n = k a n,k k -s and we have used (2.1) for dµ(u) = du 1+u 2 . Since (f n ) tends weakly to zero and θ > 0, it is easy to prove that k |a n,k | 2 k -θ goes to zero as n tends to +∞. Hence we have shown that for all ε > 0, lim sup n→+∞ I n,1 ≤ ε which implies that lim n→+∞ I n,1 = 0.

Let us turn to I n,2 . Again for arbitrary ε > 0 there exists θ ∈ (0, 1) such that Re (ϕ χ (s)) < θ implies |u χ (s)| ≤ εRe ϕ χ (s) .

Keeping the notation Ω χ = {(σ, t) ∈ [0, 1] 2 : Re (ϕ χ (s)) ∈ (0, θ)} and using Re (s) ≤ Re ϕ χ (s) /c 0 , We obtain

T ∞ Ωχ |u χ (σ + it)| 2 |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) ≤ 1 c 0 ε 2 T ∞ 1 0 1 0 |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 Re 3 ϕ χ (σ + it) dσdtdm(χ)
c 0 ,C ε 2 by the same argument as that of Theorem 5.3. On the other hand, arguing as above,

T ∞ [0,1] 2 \Ωχ |u χ (σ + it)| 2 |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) c 0 ,C T ∞ [0,1] 2 \Ωχ |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 Re 3 (ϕ χ (σ + it))dσdtdm(χ).
For a fixed χ ∈ T ∞ , we do the nonunivalent change of variables w = ϕ χ (s). Observe that we now have, for all χ ∈ T

∞ , ϕ χ ([0, 1] 2 \Ω χ ) ⊆ [θ, c 0 +C]×[-c 0 -C, c 0 +C].
We get, using again the Nevanlinna counting function N ϕχ,3 as in the proof of Theorem 5.3,

T ∞ [0,1] 2 \Ωχ |u χ (σ + it)| 2 |ϕ χ (σ + it)| 2 |f n,χ c 0 • ϕ χ (σ + it)| 2 σdσdtdm(χ) c 0 ,C T ∞ c 0 +C -c 0 -C σ 0 θ |f n,χ c 0 (σ + it)| 2 σ 3 dσdtdm(χ) c 0 ,C +∞ k=2 (log k)|a k,n | 2 k -2θ
and again it is easy to prove that this last quantity goes to zero as n goes to +∞ by weak convergence of (f n ) to zero. This concludes the proof that uC ϕ is a compact operator from A to H.

5.2.2.

Applications to connected components and compact differences. Lemmas 5.1 and 5.2, Theorems 5.3 and 5.4 lead us to the following sufficient conditions for two composition operators to be in the same component or for their difference to be compact.

Theorem 5.5. Let ϕ 0 and ϕ 1 ∈ G with char (ϕ 0 ) = char (ϕ 1 ) =: c 0 ≥ 1 and write them ϕ 0 = c 0 s + ψ 0 , ϕ 1 = c 0 s + ψ 1 . Assume moreover that there exists C > 0 such that

• |ϕ 0 -ϕ 1 | ≤ C min(Re ϕ 0 , Re ϕ 1 ); • |ψ 0 |, |ψ 1 | ≤ C; • |ϕ 0 -ϕ 1 | ≤ C. Then C ϕ 0 and C ϕ 1 belong to the same component of C(H). If moreover we assume that |ϕ 0 -ϕ 1 | = o min(Re ϕ 0 , Re ϕ 1 ) and ϕ 0 -ϕ 1 → 0 as min(Re ϕ 0 , Re ϕ 1 ) → 0 then C ϕ 0 -C ϕ 1 is compact.
We can deduce from this a result about non isolation of composition operators.

Corollary 5.6. Let ϕ = c 0 s + ψ ∈ G with positive characteristic. Assume that ψ and ψ are bounded on C + and that there exist

C > 0, k ≥ 1, t 0 ∈ R such that, for all s ∈ C + , |Im (ψ(s)) -t 0 | ≤ C Re (ψ(s)) 1/k . Then C ϕ is not isolated in C(H).
Proof. Without loss of generality, we may assume that t 0 = 0. Let us denote ϕ 0 = ϕ and let, for δ > 0, ϕ 1 = c 0 s + ψ + δψ k . We claim that, provided δ is small enough, ϕ 0 and ϕ 1 satisfy the assumptions of Theorem 5.5. For s ∈ C + , let us write

Re ϕ 1 (s) = c 0 Re (s) + Re ψ(s) + δRe (ψ k (s)). Now, Re (ψ k (s)) = k j=0 a j Re ψ(s) j Im ψ(s) k-j
for some coefficients a j . Using our assumption on Im ψ and the boundedness of Re ψ, there exist C 0 > 0 such that, for all s ∈ C + , Re (ψ k (s)) ≥ -C 0 Re (ψ(s)).

We set δ = 1/2C 0 . We get for all s

∈ C + Re ϕ 1 (s) ≥ c 0 Re (s) + 1 2 Re ψ(s) ≥ 1 2
Re ϕ 0 (s).

Therefore ϕ 1 ∈ G. Furthermore, for all s ∈ C + , with involved constants independent of s,

|ϕ 0 (s) -ϕ 1 (s)| 2 = δ 2 |ψ(s)| 2k Re ψ(s) 2k + Im ψ(s) 2k Re ψ(s) 2 min(Re ϕ 0 (s), Re ϕ 1 (s)) 2 ,
where, again, we used our assumption on Im ψ and the boundedness of Re ψ.

Corollary 5.7. Let ϕ 0 (s) = c 0 s+c 1 + d j=1 c q j q -s j be a linear symbol with char (ϕ 0 ) ≥ 1. Then C ϕ 0 is not isolated in C(H). Moreover, if ϕ 0 has unrestricted range, there exists a continuous map λ ∈

[0, δ] → ϕ λ ∈ G such that, for λ = λ , C ϕ λ -C ϕ λ is never compact.
Proof. We only have to prove the statement when C ϕ 0 is not compact, namely when ϕ 0 has unrestricted range. In that case, we may assume that c q j = -|c q j | and c 1 = d j=1 |c q j |. Since s → q -s j maps C + into D, ψ 0 maps C + into some disk in C + which is internally tangent to the imaginary axis at 0. Therefore the assumptions of Corollary 5.6 are satisfied with k = 2. Moreover, a look at the proof of Theorem 5.5 shows that we can choose a continuous path λ → ϕ λ given by

ϕ λ (s) = c 0 s + c 1 + d j=1 c q j q -s j + λ c 1 + d j=1 c q j q -s j 2
.

By Theorem 4.5, for λ = λ , C ϕ λ -C ϕ λ is never compact.

In particular, the Shapiro-Sundberg conjecture keeps false in C(H). We now show the optimality of Theorem 4.5, by proving that we cannot extend it to polynomial symbols of degree 3.

Corollary 5.8. Let ϕ 0 = c 0 s + ψ 0 ∈ G with positive characteristic. Assume that ψ 0 and ψ 0 are bounded on C + and that there exist

C > 0, k ≥ 1, t 0 ∈ R such that, for all s ∈ C + , |Im (ψ 0 (s)) -t 0 | ≤ C Re (ψ 0 (s)) 1/k . Let δ > 0 and ϕ 1 = c 0 s + ψ 0 + δψ k+1 0 . Then, for δ > 0 sufficiently small, ϕ 1 ∈ G and C ϕ 0 -C ϕ 1 is compact.
Proof. The proof follows that of Corollary 5.6. With the same computations, still assuming t 0 = 0, one now gets

|ϕ 0 (s) -ϕ 1 (s)| 2 min(Re ϕ 0 (s), Re ϕ 1 (s)) 2(k+1)/k
which is enough to ensure that |ϕ 0 -ϕ 1 | = o min(Re ϕ 0 , Re ϕ 1 ) as min(Re ϕ 0 , Re ϕ 1 ) tends to zero. We conclude by applying Theorem 5.5 since ϕ 1 -ϕ 0 = δ(k + 1)ψ 0 ψ k and since the assumptions imply that ψ 0 tends to zero as min(Re ϕ 0 , Re ϕ 1 ) tend to zero.

Example 5.9. Let ϕ 0 (s

) = s + (1 -2 -s ) and ϕ 1 (s) = s + (1 -2 -s ) + δ(1 -2 -s ) 3
for some sufficiently small δ > 0. Then C ϕ 1 -C ϕ 0 is compact whereas neither C ϕ 0 nor C ϕ 1 is compact. In particular, Theorem 4.5 is optimal.

5.3. The case of zero characteristic.

5.3.1.

Weighted composition operators from Bergman spaces to Hardy spaces. In this section, we give a sufficient condition for a weighted composition operator uC ϕ to induce a bounded, respectively compact, operator from the Bergman space A 2 1 (D) to the Hardy space H 2 (D). Here A 2 1 (D) is the set of analytic functions f on the disk such that

f 2 A 2 1 (D) := D |f (z)| 2 (1 -|z| 2 )dz < +∞.
The continuity of uC ϕ : A 2 1 (D) → H 2 (D) has been characterized in terms of Carleson measures in [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF] or using integral operators in [START_REF] Stević | Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball[END_REF]. Inspired by [START_REF] Al-Rawashdeh | Difference of composition operators on Hardy space[END_REF], we shall deduce from the results of [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF] a sufficient condition which will be easy to testify. Let us introduce some terminology. For a point ξ in the boundary of the unit disk and δ > 0, we define the Carleson window S(ξ, δ) = {z ∈ D : |z -ξ| < δ}.

Let ϕ be an analytic self-map of D and let u be a bounded analytic function on D. We define a positive Borel measure |u| 2 m 1 ϕ -1 on the closed unit disk D by assigning to each Borel set E the value:

|u| 2 m 1 ϕ -1 (E) = ϕ -1 (E) |u(ξ)| 2 dm 1 (ξ),
where we still denote by ϕ and u the radial limit functions of ϕ and u and where m 1 is the normalized Lebesgue measure on T.

The following lemma is [23, Proposition 2].

Lemma 5.10. Let ϕ be an analytic self-map of D and let u be a bounded analytic function on D not identically equal to zero. Then uC ϕ : A 2 1 → H 2 is bounded if and only if ϕ has radial limits of modulus strictly less than 1 almost everywhere and .

If sup S(ξ,δ) |u| 2 m 1 ϕ -1 (S(ξ, δ))/δ 3 goes to zero uniformly in ξ as δ tends to zero, then uC φ is a compact operator from A 2 1 to H 2 . Remark 5.11. In the statement of [START_REF] Moorhouse | Differences of composition operators, Trends in Banach spaces and operator theory[END_REF], the exponent 1/2 in the last display does not appear. Nevertheless a look at the proof as well as homogeneity considerations show that it has to appear. Corollary 5.12. Let ϕ be an analytic self-map of D which has radial limits of modulus strictly less than 1 almost everywhere and let u be a bounded analytic function on D not identically equal to zero. Then The statement about compactness follows from the second half of Corollary 5.12. )). Indeed, we apply Corollary 5.12 not to u but to (z -1) 2 u/2 which allows this better inequality. Nevertheless, since we plan to apply this result to vertical limits, with a uniform bound, the term 1 + t 2 would not be helpful to us.

5.3.2.

Applications to connected components and compact differences. Proof. We start as in the proof of Theorem 5.5 but now we choose the measure dµ = dt 1+t 2 . Taking into account that the symbols have zero characteristic, we know that for all λ, λ ∈ [0, 1],

C ϕ λ (f ) -C ϕ λ (f ) 2 ≤ |λ -λ| 2 λ λ T ∞ R |(ϕ 1 ) χ (it) -(ϕ 0 ) χ (it)| 2
×|f ((ϕ r ) χ (it))| 2 dt 1 + t 2 dm(χ)dr. Now H embeds into H 2 i (C + ) via the following inequality ( [START_REF] Brevig | Sharp norm estimates for composition operators and Hilbert-type inequalities[END_REF]): for all g ∈ H, g

H 2 i (C 1/2 ) ≤ √ 2 g H .
Therefore we just need to prove that there exists A > 0 such that, for all χ ∈ T ∞ , for all r ∈ [0, 1] and for all g ∈ H 2 i (C 1/2 ), (ϕ 1 -ϕ 0 ) χ • g • (ϕ r ) χ H 2 i (C + ) ≤ A g H 2 i (C 1/2 ) . For a fixed a > 0, the condition Re z-1/2 |1+z| 2 ≥ a is equivalent to say that z belongs to some disk. Since every disk is convex, we get that for all r ∈ [0, 1],

|ϕ 0 -ϕ 1 | ≤ C Re ϕ r -1/2 |1 + ϕ r | 2 .
as w σ → z.

Proof. We may assume that z = e and that Bψ 0 (e) = Bψ 1 (e) = 0. We set m = 2n-1 or m = 2n, following the assumptions that we have made. These assumptions imply that, looking at some w = (e iθ 1 , . . . , e iθ d ), for some constant C 1 > 0.

On the other hand, let i = 0, 1 and let us write the Taylor expansion of Bψ i at w as Bψ i (w σ ) = Bψ i (w) + d j=1 a i,j (w)(1 -p -σ j ) + F i (w, σ)

where the functions a i,j are continuous and |F i (w, σ)| ≤ C 2 σ 2 . Expanding 1 -p -σ j , this can been rewritten as Bψ i (w σ ) = Bψ i (w) + d j=1 a i,j (w) log p j σ + G i (w, σ) with |G i (w, σ)| ≤ C 3 σ 2 . Now, we know (see [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF]) that a i,j (e) ≥ 0 and a i,j 0 (e) > 0 for at least one j 0 ∈ {1, . . . , d}. From the continuity of the functions a i,j , we deduce the existence of a neighbourhood U of e in T d , of C 4 > 0 and of σ 0 > 0 such that, for all w ∈ U and for all σ ∈ [0, σ 0 ], Re Bψ i (w σ ) ≥ Re Bψ i (w) + C 4 σ

≥ C 4 σ + C 5 (θ 2n 1 + • • • + θ 2n d )
where the last inequality holds because ϕ i has a Dirichlet contact of order 2n at e. This yields the following lemma.

  (a) If c 0 = 0, then ψ(C 0 ) ⊆ C 1/2 . (b) If c 0 ≥ 1, then either ψ ≡ c 1 with Re c 1 ≥ 0 or ψ(C 0 ) ⊆ C 0 .

Definition 4 . 1 .

 41 Let F, G : D d → C + be of class C 2 and let ξ ∈ T d . We say that F and G have the same second order boundary data at ξ if F

Theorem 5 . 3 .

 53 Let ϕ ∈ G with char (ϕ) > 0 and write it ϕ = c 0 s + ψ. Let u be a Dirichlet series convergent in C + . Assume that there exists C > 0 such that • |u | ≤ C, |ψ| ≤ C; • |u| ≤ CRe ϕ. Then uC ϕ A→H c 0 ,C 1.

3 ≤ c 3 0

 33 Re 3 (w) by [6, Proposition 3], we can finally conclude that

Theorem 5 . 4 .

 54 Let ϕ ∈ G with char (ϕ) > 0 and write it ϕ = c 0 s + ψ. Let u be a Dirichlet series convergent in C + . Assume that there exists C > 0 such that • |u | ≤ C, |ψ| ≤ C; • |u| ≤ CRe ϕ; • |u | → 0 and |u| = o Re (ϕ) as Re (ϕ) → 0. Then uC ϕ is a compact operator from A to H.

= 1 δ 3 ϕ 2 × 1 δ 3 ϕ 2 × 1 δ m 1 ϕ 2 H 2

 3232122 uC ϕ is a compact operator from A 2 1 to H 2 . Proof. We simply write, for ξ ∈ T and δ > 0, |u| 2 m 1 ϕ -1 (S(ξ, δ)) δ 3 -1 (S(ξ,δ))|u| 2 dm 1 ≤ sup ξ 0 ∈T |u(ξ 0 )| 1 -|ϕ(ξ 0 )| -1 (S(ξ,δ)) (1 -|ϕ|) 2 dm 1 ≤ sup ξ 0 ∈T |u(ξ 0 )| 1 -|ϕ(ξ 0 )| -1 (S(ξ, δ)) .It follows from the Carleson measure characterisation of the continuity of composition operators onH 2 (D) that 1 δ m 1 ϕ -1 (S(ξ, δ)) C ϕ →H 2 ≤ 1+|ϕ(0)| 1-|ϕ(0)| .We translate this statement to get a result on the invariant Hardy space H 2 i (C + ). Lemma 5.13. Let φ be an analytic self-map of C + which has radial limits of positive real part almost everywhere and let U be a bounded analytic function on C + not identically equal to zero. Then for all F ∈ H 2 i (C + ),U • F • φ H 2 i (C + ) |1 + φ(1)| 2 Re φ(1) 1/2 sup t∈R |U (it)| × |1 + φ(it)| 2 Re φ(it) F H 2 i (C + ) . If, moreover, sup |U (it)| × |1 + φ(it)| 2 Re φ(it) : Re φ(it) |1 + φ(it)| 2 ≤ δ δ→0 --→ 0 then the operator F → U • F • φ is a compact operator of H 2 i (C + ). Proof. Let ω be the Cayley map, ω(z) = 1+z 1-z and let us set f = F • ω, u = U • ω, ϕ = ω -1 • φ • ω. Observe that U F • φ = g • ω -1 where g = u × (ω -1 ) • ω × f • ϕ = (z -1) 2 2 u × f • ϕ.A small computation shows thatsup ξ∈T |u(ξ)| 1 -|ϕ(ξ)| 2 ≈ sup t∈R |U (it)| × |1 + φ(it)| 2Re φ(it)and 1 + |ϕ(0)| 1 -|ϕ(0)| ≈ |1 + φ(1)| 2 Re φ(1) . Therefore U • F • φ H 2 i (C + ) = g H 2 (D) |1 + φ(1)| 2 Re φ(1) )| × |1 + φ(it)| 2 Re φ(it) F H 2 i (C + ) .

Remark 5 . 14 .

 514 We could improve the statement of Lemma 5.13 by replacing everywhere the term|U (it)| × |1 + φ(it)| 2 /Re φ(it) by |U (it)| × |1 + φ(it)| 2 /(Re φ(it) × (1 + t 2

Theorem 5 . 15 . 2 |1+ϕ 1 | 2 . 2 |1+ϕ 1 | 2 as 2 |1+ϕ 1 | 2 tends

 515212212212 Let ϕ 0 , ϕ 1 ∈ G with char (ϕ 0 ) = char (ϕ 1 ) = 0. Assume that there exists C > 0 such that |ϕ 0 -ϕ 1 | ≤ C min Re ϕ 0 -1/2 |1+ϕ 0 | 2 , Re ϕ 1 -1/Then C ϕ 0 and C ϕ 1 belong to the same component of C(H). If, moreover, |ϕ 0 -ϕ 1 | = o min Re ϕ 0 -1/2 |1+ϕ 0 | 2 , Re ϕ 1 -1/min Re ϕ 0 -1/2 |1+ϕ 0 | 2 , Re ϕ 1 -1/to zero, then C ϕ 0 -C ϕ 1 is compact.

|Bψ 0

 0 (w σ ) -Bψ 1 (w σ )| = |Bψ 0 (w σ ) -1 + 1 -Bψ 1 (w σ )| = |α|=m+1 a α (1 -2 -σ e iθ 1 ) α 1 • • • (1 -p -σ d e iθ d ) α d + o( w σ -e m+1 ) ≤ C 1 σ m+1 + |θ 1 | m+1 + • • • + |θ d | m+1

Lemma 5 . 20 .

 520 Let ϕ 0 , ϕ 1 ∈ G be Dirichlet polynomial symbols with char (ϕ 0 ) = char (ϕ 1 ) ≥ 1. Assume that Γ(Bψ 0 ) = Γ(Bψ 1 ) and that, for all z ∈ Γ(Bψ 0 ), there exists n ∈ N such that• Bψ 0 (z) = Bψ 1 (z);• ϕ 0 and ϕ 1 have a Dirichlet contact of order 2n at z;• for |α| ≤ 2n -1, ∂ α Bψ 0 (z) = ∂ α Bψ 1 (z).Then there exists C > 0 such that, for all w ∈ T d and all σ ≥ 0,|Bψ 0 (w σ ) -Bψ 1 (w σ )| ≤ C min Re Bψ 0 (w σ ), Re Bψ 1 (w σ ) .Moreover, if we assume that ∂ α Bψ 0 (z) = ∂ α Bψ 1 (z) for |α| ≤ 2n, then |Bψ 0 (w σ ) -Bψ 1 (w σ )| = o min Re Bψ 0 (w σ ), Re Bψ 1 (w σ )

  ). Then C ϕ 0 and C ϕ 1 do not belong to the same component of C(H). Corollary 3.11 already gives interesting examples of composition operators that do not belong to the same component of C(H). However, it does not help us to solve the following simple problem: does C ϕ 0 and C ϕ 1 belong to the same component of C(H) where

Acknowledgment. F. Bayart was supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front), M. Wang and X. Yao were supported by National Science Foundation of China (Nos. 12171373, 11771340, 11701434).

By vertical translation, we also have

Moreover, since ϕ 0 and ϕ 1 converge uniformly on C 1/2 , there exists some compact subset K ⊆ C 1/2 such that, for all χ ∈ T ∞ and all r > 0, (ϕ r ) χ (1) belongs to K. Therefore, Lemma 5. [START_REF] Brevig | A mean counting function for Dirichlet series and compact composition operators[END_REF] gives the desired result. The proof of the compactness of C ϕ 0 -C ϕ 1 follows the same lines and is left to the reader.

We then get a result similar to Corollary 5.6 for composition operators with zero characteristic.

Corollary 5.16. Let ϕ ∈ G with zero characteristic and such that |ϕ| is bounded. Assume that there exist

Proof. We set ϕ 0 = ϕ and let, for δ > 0,

As in the proof of Corollary 5.6, ϕ 1 ∈ G provided δ is small enough. Moreover, the assumptions of Corollary 5.16 are satisfied since

since |ϕ| is bounded.

We deduce from this result that the Shapiro-Sundberg conjecture remains false for composition operators with zero characteristic.

Example 5.17. Let ϕ 0 (s) = 1 2

Proof. The first half of the statement comes from Corollary 5.16 (and its proof). The second half is a consequence of Theorem 4.5 since neither ϕ 0 nor ϕ 1 is compact by [7, Theorem 2]. 5.4. Order of contact, connected components and compact differences. In this final section, we show how the conditions of Theorem 5.5 can be satisfied from conditions on the derivatives of Bψ. We need a notion of order of contact, which is maybe less natural than in one variable (see [START_REF] Kriete | Linear relations in the Calkin algebra for composition operators[END_REF]).

Let ϕ ∈ G be a Dirichlet polynomial symbol with char (ϕ) ≥ 1. Let d be the smallest integer such that

Recall also that Γ(Bψ) is defined by Γ(Bψ) = {z ∈ T d : Re Bψ(z) = 0} (this set is nonempty provided ϕ has unrestricted range). Let z = (e iθ 0 1 , . . . , e iθ 0 d ) ∈ Γ(Bψ). Then J ϕ attains its minimum at (θ 0 1 , . . . , θ 0 d ). Definition 5.18. We say that ϕ has a Dirichlet contact of order 2n at z = (e iθ 0 1 , . . . , e iθ 0 d ) provided there exists c > 0 and a neighbourhood U of (θ 0 1 , . . . , θ 0 d ) in R d such that, for all (θ 1 , . . . , θ d ) ∈ U,

Let us explain this terminology. Without loss of generality, we may assume that θ 0 1 = • • • = θ 0 d = 0 and Bψ(1, . . . , 1) = 0. The Taylor expansion of Bψ at z may be written

with each a j ≥ 0 (see [START_REF] Bayart | Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series[END_REF]). In particular, there exists a neighbourhood V of (0, . . . , 0) in R d such that, for all (θ 1 , . . . , θ d ) ∈ V, |Im Bψ(e iθ 1 , . . . , e

|Im Bψ(e iθ 1 , . . . , e iθ d )| 2n Re Bψ(e iθ 1 , . . . , e iθ d ).

Hence, if ϕ has a Dirichlet contact of order 2n at z, the Bohr lift Bψ has a contact of order 2n at z.

For w ∈ T d and σ > 0, we denote by

• ϕ 0 and ϕ 1 have a Dirichlet contact of order 2n at z;

Then there exist a neighbourhood U of z in T d , σ 0 > 0 and C > 0 such that, for all w ∈ U, for all σ ∈ [0, σ 0 ],

Moreover, if we assume that ∂ α Bψ 0 (z) = ∂ α Bψ 1 (z) for |α| ≤ 2n, then |Bψ 0 (w σ ) -Bψ 1 (w σ )| = o min Re Bψ 0 (w σ ), Re Bψ 1 (w σ ) as min Re Bψ 0 (w σ ), Re Bψ 1 (w σ ) goes to zero.

Proof. This follows from the previous lemma and a compactness argument.

Corollary 5.21. Let ϕ 0 , ϕ 1 ∈ G be Dirichlet polynomial symbols with char (ϕ 0 ) = char (ϕ 1 ) ≥ 1. Assume that Γ(Bψ 0 ) = Γ(Bψ 1 ) and that, for all z ∈ Γ(Bψ 0 ), there exists n ∈ N such that

• Bψ 0 (z) = Bψ 1 (z);

• ϕ 0 and ϕ 1 have a Dirichlet contact of order 2n at z;

Proof. The previous lemma implies that the assumptions of Theorem 5.5 are satisfied.

Combining Theorem 4.4 and the previous corollary, we finally get: Theorem 5.22. Let ϕ 0 , ϕ 1 ∈ G be Dirichlet polynomial symbols with char (ϕ 0 ) = char (ϕ 1 ) ≥ 1. Assume that for all z ∈ Γ(Bψ 0 ) (resp. z ∈ Γ(B(ψ 1 )), ϕ 0 (resp. ϕ 1 ) has Dirichlet order of contact 2 at z. Then the following assertions are equivalent:

(i) C ϕ 0 -C ϕ 1 is compact.

(ii) Γ(Bψ 0 ) = Γ(Bψ 1 ) and, for all z ∈ Γ(Bψ 0 ), Bψ 0 (z) = Bψ 1 (z), ∂ α Bψ 0 (z) = ∂ α Bψ 1 (z) for |α| ≤ 2.