Aleksandra Mozdzanowska

R John Hansman

Jonathan Histon

Daniel Delahaye

EMERGENCE OF REGIONAL JETS AND THE IMPLICATIONS ON AIR TRAFFIC MANAGEMENT

Airlines are increasingly using regional jets to better match aircraft size to high value, but limited demand markets.

This has been especially important following increased financial pressure on the industry after September 11 th 2001. The increase in regional jets represents a significant change from traditional air traffic patterns. To investigate the possible impacts of this change, this study analyzed the emerging flight patterns and performance of regional jets compared to traditional jets and turboprops. In addition, a comparison between regional jet flight patterns in the United States and Europe was conducted.

Regional jet operations generally cluster in the regions with high traditional jet operation density, implying a high level of interaction between the two aircraft types. The regional jets were observed to fly shorter routes than traditional jets, with few transcontinental flights. However, the gap between regional and narrow body traditional jet stage lengths appears to be closing. In addition, regional jets were observed to exhibit lower climb rates than traditional jets, which may impact air traffic control handling and sector design. It was also observed that regional jets cruise at lower altitudes than traditional jets possibly due to their shorter flight routes. Finally, it was observed that regional jets cruise at a lower Mach number than traditional jets, except on specific high density routes where the regional jets are either slowing down the traditional jet traffic or flying above their optimum cruise speed. Since the composition and utilization of the national fleet is changing, this will pose potential problems for air traffic management. In particular, it may cause serious congestion issues when demand increases during an economic recovery.

Introduction

One of the significant emerging changes to the national air space system is the number of regional jets, which has been increasing at an exponential rate.

The rapid emergence of regional jets can be seen in Figure 1, which shows FAA registration data plotted between the third quarter of 1993 and the third quarter of 2002, for some of the regional jets commonly flown in the United States. The figure shows that the growth in regional jets is exponential, with CRJ2 and E145 registration data increasing the fastest. Airlines are moving to regional jet operations because the smaller aircraft size allows them to better match the aircraft to high value markets, and the faster turnaround times allow for a higher frequency of service. Regional jets have similar flight profiles, as traditional jets. As a result, the two jet types pose similar demands on airports and airspace. Currently the result of these demands is not fully known, and as a result, the constraints that will emerge are also unknown. This paper analyzes those demands and their possible implications on the air traffic system by studying the national flight patterns as well as the performance characteristics of regional jets, traditional jets, and turboprops.

Methodology

In order to visualize the emerging trends in the United States, flight data from the Aircraft Situational Display to Industry (ASDI) feed was used. This data is compiled by the Volpe center from the FAA's Enhanced Traffic Management System (ETMS). This study used data for all aircraft that had a take off time between midnight December 19 th , 2002 GMT and midnight December 20 th , 2002 GMT. While only one day of data is presented here, data starting in April 2002 has been analyzed and has shown similar results. This set of data was broken down into four categories based on aircraft type: the narrow body traditional jets, the wide body traditional jets, the turboprops, and the regional jets. The list of regional jets, regional airlines, and their codes share partners can be seen in Table 1 [START_REF]Aviation Week and Space Technology[END_REF].

In order to conduct a comparison between regional jet operations in the U.S and in Europe simulated track data of European flights was used. The data was created based on filed flight plans for December 19 th 2002. A technique was developed to generate density maps from the traffic data. An example of a density map is shown in Figure 2. The map shows the number of aircraft which appeared over a specific area during the course of an input time bound. The plot was generated by dividing the map into a grid and counting the number of flights whose paths intersected each grid. The grid was then colored based on the number of flights that flew through it, creating an effective density map. It should be noted that when the size of the grid is changed, the resulting density values may be altered.

Regional Jet Operations

Growth of Regional Jets

The growth of regional jet networks is shown in Figures 3, 4, and5. These figures show the scheduled regional jet flights in 1998, 2000, and 2002 respectively. The data was obtained from the OAG and a straight line was drawn between the origin and destination creating a network map. It is clear from the figures that the number of regional jets increases with time, and it can also be seen that most of the growth occurs in the north-eastern part of the United States.

Scope Clause Restrictions

While the number of regional jets has been increasing at a rapid rate for the past five years, a major obstacle to the growth of regional jets has been posed by labor scope clauses. Scope clauses result from labor contracts made with airline pilots and typically limit the number and size of regional jets that an airline can own or operate. Currently it is less expensive for an airline to fly regional jets because the pilots are typically paid much less than the mainline crews. Given that airlines have been facing serious financial problems since 9/11/2001, many are renegotiating their existing pilot contracts. As a result, the effect of scope clauses is likely to change. At this point it is unclear if changes in scope clauses will result in an increase or decrease in the rate of regional jet operations. Currently only USAirways and United have reached agreements, and both could leverage bankruptcy during the negotiations. However, if the economy does not recover rapidly other carriers may face bankruptcy and force their pilots to agree to more lenient scope clause restrictions.

In addition to limiting growth, scope clauses can help shape the regional jet fleet that an airline owns, since regional jets have to fit within the scope clause limitations. For example, in the fall of 2001, Continental Express had no specific limitation on the number of regional jets. However, the mainline scope clause stated that if the company owned jets with over 59 seats, both the regional and mainline pilots would fly the aircraft. Table 2 shows the number, type, and size of all regional jets owned by Continental Express as of December 2002 [3]. It can be seen that the airline owns only regional jets with less then 50 seats. This is the largest regional jet available with less then 59 seats; the next model has 70 sets. It is possible that Continental Express could not identify a market for larger aircraft, but it is more likely that the increased crew costs made the larger regional jets less attractive.

Utilization of Regional Jets

In addition to the rapid growth, the utilization of regional jets is changing. Figure 6 shows the utilization of regional jets introduced into the national air space system between 1992 and 2001 [4]. For that period, it can be seen that a majority of the incoming jets created new routes, and that replacement of turboprops and traditional jets, as well as supplementing traditional jet routes had a smaller impact. This trend seems to have changed after 9/11/2001. It is expected that now most regional jets are replacing and supplementing traditional jet routes, but more data is needed before this can be firmly concluded. However, it is consistent with the similar operation patterns between regional jets and traditional jets, shown throughout this paper

Regional Jet Flight Patterns

In order to study the effects of the dramatic increase of regional jets it is illustrative to look at their flight patterns over the domestic United States compared to the patterns of traditional jets and turboprops. A U.S. density map for each aircraft category is shown in Figures 7,8, 9, and 10. It can be seen from Figure 7 that the regional jets appear in high numbers in the north-east U.S., and that their operations are focused around the hubs forming networks in much the same way as the narrow body traditional jets. While the flight distances are significant there are very few transcontinental flights.

Figure 8 shows that the narrow body traditional jet flights cover all of the United States and are focused at the hubs. Their largest concentration appears in the north-east, but their presence in the mid-west is also significant. The similarity in operating patterns implies that the interactions between regional and narrow body traditional jets will be the most noticeable at the hubs and in the north-east U.S. Although the narrow body traditional jets are concentrated in the same area of the country as regional jets, it can be seen from the map that they have a significant number of transcontinental flights. Figure 9 show the wide body traditional jet density. The map shows that the wide body traditional jets have a higher density in the mid-west and a lower density in the south-east. They are not as hub focused as the narrow body traditional jets or the regional jets. The wide body traditional jets have a high number of transcontinental and international flights. Finally, Figure 10 shows that turboprops are also concentrated in the north-east. They are focused around hubs and other larger airports, and have no transcontinental flights and very few flights that connect major airports. These short flight distances are due to the fact that turboprops do not have the capability to fly long routes. As a result, they are mostly used to feed larger airports. 3 that the European airspace handles fewer flights than the U.S airspace. Europe also has a higher percentage of traditional jets flights than the United States, but the United States has more regional jets and significantly more turboprops. This is most likely due to the fact that Europe has an extensive network of trains, which are used for short trips instead of the turboprops. The relative route maps of all the aircraft types in both Europe and the United States can be seen in Figures 11 through 18. The wide body and narrow body jet track patterns look the same between the United States and Europe. In both cases the wide body traditional jets provide international flights, and oceanic flights. The narrow body jets provide transcontinental flights in the United States and international (transcontinental) flights in Europe. The major difference between the United States and Europe can be seen when comparing the regional jets and turboprop operating patterns. In the United States the regional jet flight patterns are very similar to the narrow body jet patterns, and the turboprops can be seen to closely center on the hubs. In Europe the regional jet and turboprop patterns are extremely similar, both aircraft fly short routes, and with less concentration around hubs than is observed in the US operating patterns.

Stage Length Differences

The analysis of the density maps prompted a closer examination of flight distances in the U.S data. Figure 19 shows a distance histogram analyzed by aircraft category.

The flight distances were calculated as the great circle distance between the first and last available ETMS record for each flight. The plot shows that turboprops fly short distances, with an average stage length of 187 statute miles, and are not observed to fly routes longer than about 800 statute miles.

The regional jet operations outnumber the narrow body traditional jet operations for flights below 650 statute miles. However, for flights above that value the regional jet frequency tapers rapidly, and no regional jets are observed on flights longer then about 1200 statute miles. This results in an average stage length of 394 statute miles for regional jets, and 710 statute miles for narrow body traditional jets. Given that the number of regional jets is increasing, and that some of these jets are replacing traditional jet routes, it is expected that the gap between regional and traditional jets above 650 statute miles will close with time. This prediction is supported by comparing the current average stage length of regional jets to one calculated in 1998, which was 375 statute miles [START_REF] William | The Impacts of Regional Jets on Congestion in the NAS[END_REF]. Finally, the wide body traditional jets have a similar distribution of flight distances as the narrow body traditional jets. However, they have a higher frequency of flights above 1300 statute miles. As a result, the average stage length for wide body traditional jets is 897 statute miles.

Figure 19: Stage Length Histogram

To examine a specific example of stage length differences, the flight paths of aircraft leaving Dallas Forth Worth were plotted and colored by aircraft category. The results, shown in Figure 20, demonstrate the increase in stage length from turboprops, to regional jets, to narrow body traditional jets, to wide body traditional jets. In addition it can be seen that most of the regional jets flights are to the east, which is consistent with the results from the density map.

Implications of Regional Jet Performance

Given the rapid growth of regional jets, and their high level of interaction with other aircraft types, it is critical to analyze the implications of these interactions.

The observed performance of regional jets in the U.S. data was compared to narrow and wide body traditional jets and turboprops at the airport and terminal areas as well as during cruise.

Implications at the Airport and Terminal Area

In the airport and terminal areas, the main issues resulting from increased regional jet operations will be congestion related. Regional jets typically need to use the same runways as traditional jets, whereas turboprops often use shorter ones. As the number of regional jets increases, they will be competing for runway space, and may exacerbate congestion problems as the number of operations per person will increase.

In addition to runway constrains, local airspace limitations may be exacerbated by regional jet operations. Figure 21 shows the flight tracks for departures from DFW on the 6 th of February 2003 between 0000 and 0500 GMT. The black lines indicate turboprop trajectories, and the blue and green lines represent regional and traditional jet trajectories. It can be seen that regional and traditional jets align on the same tracks, while turboprops fly on separate routes. As turboprops are replaced by regional jets, an increase in congestion on the jet arrival and departure routes is expected.

Figure 21: Flight Tracks for Aircraft Leaving DFW

In addition to congestion issues due to an increase in operations, congestion and delay issues may also appear due to climb performance differences between regional and traditional jets. Figure 22, shows the altitude vs. time plot for flights between CLE and ORD. It can be seen from the plot that both the CRJ and ERJ climb slower then the Boeing or Airbus jets. The ERJ in particular stands out as having a slower climb rate. This data represents only one example, and this phenomenon is difficult to study on a system wide basis; however, a number of routes have been studied and have produced similar results. The slower climb rate of regional jets increases the range of aircraft performance that controllers must manage in order to adhere to proper separation standards. Slower climb rates can change how regional jets interact with the sector structure. Many of the irregular sector shapes found in the current air traffic management system are a direct consequence of modifications targeted at reducing the number of sector boundary crossings. Boundary crossings are avoided due to the increased radio communications and coordination workload associated with each transition. Figure 23 shows a close up of Figure 22 overlaid with a hypothetical sector structure. It can be seen from the figure how the slower climb performance of regional jets could increase the number of sector boundary crossings. This implies that some departure sectors may need to be redesigned in order to accommodate this new range of aircraft performance. The dotted line in the figure shows the proposed new boundary for the lower sector.

Implications at Cruise

In order to identify potential cruise implications, high altitude cruise performance for all four aircraft types was analyzed. Figure 24 shows a normalized histogram of the primary altitude during cruise for regional jets, narrow body traditional jets, wide body traditional jets, and turboprops. The plot shows that turboprops cruise at the lowest altitude and interact only slightly with regional and traditional jets. The fact that turboprops cruise at a lower altitude is not surprising since this is one of their design characteristics. The wide body traditional jets cruise at the highest altitude, and also have a small level of interaction with the regional jets. The regional jets cruise at a lower altitude then the narrow or wide body traditional jets, but have a high level of interaction with narrow body traditional jets between 25,000 and 32,000 feet. At this time it is not clear from the data why the regional jets cruise at the lower altitude. One possible explanation is the shorter stage lengths of regional jets. The aircraft may be assigned to fly at lower altitudes since they are likely to land before a traditional jet. This connection between stage length and flight altitude is shown in Figure 25. Figure 26 shows the average speeds during cruise for regional jets, narrow body traditional jets, wide body traditional jets, and turboprops. It can be seen that the turboprops fly significantly slower then the other aircraft types and the wide body traditional jets fly slightly faster then the other aircraft types. Once again the regional jets and narrow body traditional jets have a high level of overlap in their performance, with the regional jets being the slower of the two. The speed difference is another possible explanation for the lower cruise altitudes of regional jets compared to traditional jets. If the regional jets fly too slow they may be relegated to lower altitudes to not slow down the rest of the traffic and cause congestion. However, it is difficult to determine what is cause and effect.

Conclusion

Airlines are increasingly using regional jets to better match aircraft size to high value demand markets. This has been especially important due to the increased pressure on the industry following September 11 th 2001. The increase in regional jets represents a significant change from traditional air traffic patterns. To investigate the possible impacts of this change, this study analyzed the emerging flight patterns and performance, in the context of air traffic management, of regional jets compared to traditional jets and turboprops. It was found that regional jet operations cluster in the same areas as traditional jets implying a high level of interaction between the two aircraft types. The regional jets were observed to fly shorter routes then traditional jets, with few transcontinental flights. However, it was observed that this gap is closing. An increase in regional jet operations may cause delays at airports, because regional jets compete for the same runway and local airspace resources as traditional jets. Regional jets were observed to exhibit lower climb rates then traditional jets, which may have implications for air traffic control handling and sector design. In addition, it was observed that regional jets cruise at lower altitudes then traditional jets; however, the reasons for this difference are not clear. Finally, it was observed that regional jets cruise at a lower Mach number then traditional jets, except on specific high density routes where flow restrictions reduce the cruise speeds of all aircraft.

Since the composition and utilization of the national fleet is changing this implies that the air traffic management system will need to evolve.

Figure 1 :

 1 Figure 1: FAA Registration Data for Regional Jets.

Figure 2 :

 2 Figure 2: Density Map of all Flights between December 19th and 20th, 2002

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Regional Jet Network Map in 1998 (Courtesy of Ryan Tam)

Figure 6 :

 6 Figure 6: Route Changes between 1992 and 2001 Caused by Emerging Regional Jets [4]

Figure

  Figure 7: Regional Jet Density

Figure

  Figure 11: European Wide Body Traditional Jet Flights Figure 12: U.S. Wide Body Traditional Jet Flights

Figure 20 :

 20 Figure 20: Trajectories for DFW Departing Flights (turboprops in black, regional jets in red, narrow body jets in yellow, wide body jets in blue)

Figure 22 :

 22 Figure 22: Altitude vs. Time for Flights from CLE to ORD

Figure 23 :

 23 Figure 23: Effect of Climb Rates on Sector Structure

Figure 24 :

 24 Figure 24: Normalized Histogram of Cruise Altitudes

Figure 26 :

 26 Figure 26: Normalized Histogram of Average Cruise Speeds

Table 1 : Regional Jets and their Corresponding Regional Carriers and Code Share Partners

 1

	Aircraft Type	Regional Carrier	Code Share Carrier
		American Eagle	American
	E135	Continental Express	Continental
		Republic	America West, Delta, USAirways
		American Eagle	American
		Continental Express	Continental
	E145	Mesa	America West, Frontier, USAirways
		Republic	America West, Delta, USAirways
		Trans State	American, US Airways
	CRJ1	Comair Sky West	Delta Delta, United
		Air Wisconsin	Air Tran, United
	RRJ2	Atlantic Southeast Mesa	Delta America West, Frontier, USAirways
		Sky West	Delta, United
		American Eagle	American
		Atlantic Southeast	Delta
	CRJ7	Comair	Delta
		Horizon	Alaska, Northwest
		Mesa	America West, Frontier, USAirways
	BA46	Air Wisconsin	Air Tran, United

Table 2 : Continental Express RJ Fleet as of December 2002

 2

	Aircraft	Number	Number	Restrictions
	Type	of Aircraft	of Seats
	ERJ 135	30	37	No
	ERJ 145	140	50	restrictions
	ERJ 145	8	50	on aircraft
	XR			with under
				50 seats

Table 3 : European vs. U.S. Aircraft Category Breakdowns for all Flights during a 24 Hour Period

 3

	Aircraft	Total	Percent	Total	Percent
	category	flights	of total	flight	of total
		in	flights	s in	flights
		Europe	in	the	in the
			Europe	USA	USA
	Regional	2397	13%	6783	19%
	jets
	Tradition	12268	65%	17766 50%
	al jets
	(narrow
	body)
	Tradition	2353	12%	3373	10%
	al jets
	(wide
	body)
	Turbopro	1936	10%	7378	21%
	ps

Acknowledgements

The authors would like to thank Professor David Gifford at the Laboratory for Computer Science at MIT for archiving the ASDI feed and making it available to the ICAT laboratory, Ryan Tam for providing plots of OAG data, Ian Garrick-Bethell for providing a plot of the regional jet registration data, and Emil Sit for proofreading drafts of the paper.


Maxime Bailleul

Composition operators on weighted Bergman spaces of Dirichlet series

Keywords: Mathematics Subject Classication, Primary: 47B33 -Secondary: 46E15, Keywords, weighted Bergman spaces -Carleson measure -composition operator -Dirichlet seriesessential norm -generalized Nevanlinna counting function

We study boundedness and compactness of composition operators on weighted Bergman spaces of Dirichlet series. Particularly, we obtain in some specic cases, upper and lower bounds of the essential norm of these operators and a criterion of compactness on classicals weighted Bergman spaces. Moreover, a sucient condition of compactness is obtained using the notion of Carleson's measure.

Introduction

In [START_REF] Hedenmalm | A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1)[END_REF], the authors dened the Hardy space H2 of Dirichlet series with square-summable coecients. Thanks to the Cauchy-Schwarz inequality, it is easy to see that H 2 is a space of analytic functions on C 1 2 := {s ∈ C, (s) > 1 2 }

and that this domain is maximal. F. Bayart introduced in [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] the more general class of Hardy spaces of Dirichlet series H p (1 ≤ p < +∞).

In [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], the bounded composition operators on H 2 are characterized, that is to say the holomorphic functions Φ : C 1 Theorem ( [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], [START_REF] Queélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF]). The function Φ determines a bounded composition operator on H 2 if and only:

Φ(s) = c 0 s + ϕ(s)
where c 0 is a nonnegative integer, ϕ ∈ D and ϕ converges uniformly in C ε for every ε > 0 and has the following properties:

a) If c 0 ≥ 1, ϕ(C + ) ⊂ C + . b) If c 0 = 0, ϕ(C + ) ⊂ C 1/2 . Moreover, when c 0 ≥ 1, C Φ is a contraction.
This theorem is a strengthening of the original result (see [START_REF] Queélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF],Th3.1). In [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF], F. Bayart proved that this result is true on the space H p when p ≥ 1 and c 0 = 0. When c 0 = 0 the condition is necessary for every p ≥ 1 and sucient if p is even. The goal of this paper is to study composition operators on the weighted Bergman A p µ spaces of Dirichlet series dened in [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]. In order to dene them, we need to recall the Bohr's point of view principle and the denition of the Hardy spaces H p .

Let n ≥ 2 be an integer, we shall denote e n the function dened by e n (z) = n -z . The integer n can be written (in a unique way) as a product of prime numbers n = p α1 1 • • p α k k where p 1 = 2, p 2 = 3 etc . . . If s is a complex number, we consider z = (p -s 1 , p -s 2 , . . . ). let f be a Dirichlet series of the following form

f = +∞ n=1 a n e n , (1)
then f (s) = +∞ n=1 a n (p -s 1 ) α1 • • (p -s k ) α k = +∞ n=1 a n z α1 1 • • z α k k .
So we can see a Dirichlet series as a Fourier series on the innite-dimensional polytorus

T ∞ = {(z 1 , z 2 , • • • ), |z i | = 1, ∀i ≥ 1}.
We shall denote this Fourier series D(f ).

Let us x now p ≥ 1, the space H p (T ∞ ) is the closure of the set of analytic polynomials with respect to the norm of L p (T ∞ , m) where m is the normalized Haar measure on T ∞ (see [START_REF] Cole | Representing measures and Hardy spaces for the innite polydisk algebra[END_REF] for more details). Let f be a Dirichlet polynomial, by the Bohr's point of view D(f ) is an analytic polynomial on T ∞ and by denition f H p := D(f ) H p (T ∞ ) . The space H p is dened by taking the closure of Dirichlet polynomials with respect to this norm. Consequently H p and H p (T ∞ ) are isometrically isomorphic.

For σ > 0, f σ will be the translate of f by σ, ie f σ (s) := f (σ + s). We shall denote by P the set of Dirichlet polynomials.

Let p ≥ 1, P ∈ P and let µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ). Then A p µ is the completion of P with respect to this norm. When dµ(σ) = 2e -2σ dσ, these spaces are simply denoted by A p . It is shown in [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF] that they are spaces of convergent Dirichet series on C 1/2 .

In section 2, we recall some facts about Dirichlet series and precise some notations.

In section 3, we study the boundedness of C Φ on A p µ . For Φ(s) = c 0 s + ϕ(s) with c 0 ≥ 1 and ϕ ∈ D, Φ induces a bounded composition operator from A p µ to A p µ if and only if ϕ converges uniformly on C ε for every ε > 0 and ϕ(C + ) ⊂ C + . When c 0 = 0, we give sucient and necessary conditions.

In section 4, we dene generalized Nevanlinna counting function in order to obtain some estimates of the essential norm of C Φ on H 2 and A 2 µ . Particularly we obtain a criterion of compactness of C Φ in some specic cases.

In section 5, we obtain a majorization of the generalized Nevanlinna counting function with help of the associated Carleson function (see denition in section 5) and then we get other sucient conditions of compactness of C Φ .

Background material

Let f be a Dirichlet series of form [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]. We do not recall the denition of abscissa of simple (resp. absolute) convergence denoted by σ c (resp. σ a ), see [START_REF] Queélec | Diophantine Approximation and Dirichlet Series[END_REF] or [START_REF] Tenenbaum | Introduction a la théorie analytique et probabiliste des nombres[END_REF] for more details. We shall need the two other following abscissas:

σ u (f ) = inf{a | The series (1) is uniformly convergent for (s) > a} = abscissa of uniform convergence of f. σ b (f ) = inf{a | the function f has an analytic, bounded extension for (s) > a} = abscissa of boundedness of f. It is easy to see that σ c (f ) ≤ σ u (f ) ≤ σ a (f ).
An important result is that σ u (f ) and σ b (f ) coincide: this is the Bohr's theorem (see [START_REF] Bohr | Lesung des absoluten konvergenzproblems einer allgemeinen klasse dirichletscher reihen[END_REF]), it is really important for the study of H ∞ , the algebra of bounded Dirichlet series on the right half-plane C + (see [START_REF] Maurizi | Some remarks on the algebra of bounded Dirichlet series[END_REF]). We shall denote by • ∞ the norm on this space:

f ∞ := sup (s)>0 |f (s)|.
The innite-dimensional polytorus T ∞ can be identied with the group of complex-valued characters χ on the positive integers which satisfy the following properties

|χ(n)| = 1 ∀n ≥ 1, χ(nm) = χ(n) χ(m) ∀n, m ≥ 1.
To obtain this identication for χ = (χ 1 , χ 2 , . . . ) ∈ T ∞ , it suces to dene χ on the prime numbers by χ(p i ) = χ i and use multiplicativity.

Let f be a Dirichlet series absolutely convergent in a half-plane. For any sequences (τ n ) ⊂ R, we consider vertical translations of f :

(f τn (s)) := (f (s + iτ n )).
By Montel's theorem, this sequence is a normal family in the half-plane of absolute convergence of f and so there exists a convergent subsequence, say (f τn k ), such that f τn k converges uniformly on compact subsets of the domain of absolute convergence of f to a limit function f . We say that f is a vertical limit of f .

Proposition ([10]). Let f be a Dirichlet series of the form (1), absolutely convergent in a half-plane. The vertical limit functions of f are exactly the functions of the form

f χ = n≥1 a n χ(n) e n where χ ∈ T ∞ .
When f ∈ A p µ , the vertical limits have good properties:

Proposition ([1]). Let p ≥ 1 and f ∈ A p µ . For almost all χ ∈ T ∞ , f χ converges on C + . Let Φ(s) = c 0 s + ϕ(s) where c 0 is a nonnegative integer and ϕ ∈ D. For τ ∈ R, we dene Φ τ by Φ τ (s) = c 0 s + ϕ(s + iτ ), then if χ ∈ T ∞ the function Φ χ dened by Φ χ (s) = c 0 s + ϕ χ (s)
is a vertical limit of the functions (Φ τ ) τ ∈R .

In section 4, we shall consider the Hilbert spaces of Dirichlet series H 2 and A 2 µ where µ is a probability measure on (0, +∞) such that 0 ∈ Supp(µ). Moreover we shall assume that dµ(σ) = h(σ)dσ where h is a nonnegative function such that h L 1 ((0,+∞)) = 1. We see easily that for an element of A 2 µ with form (1):

f A 2 µ = +∞ n=1 |a n | 2 w h (n) 1/2
where

w h (n) = +∞ 0 n -2σ h(σ)dσ ∀n ≥ 1.
Examples. We follow notations from [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]: let α > -1, we denote µ α the probability measure dened on (0, +∞) by

dµ α (σ) = 2 α+1 Γ(α + 1)
σ α exp(-2σ)dσ.

We denote this space A 2 α instead of A 2 µα and the corresponding weight by (w α n ).

Then for every n ≥ 1,

w α n = 1 (log(n) + 1) α+1 .
We shall denote (e µ n ) n≥1 the orthonormal basis of

Boundedness of composition operators

In the sequel µ will be a probability measure on (0, +∞) such that 0 ∈ Supp(µ) (as in [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]). Theorem 1. Let Φ : C 1 2 → C 1 2 be an analytic function of the form Φ(s) = c 0 s + ϕ(s) where c 0 ≥ 1 and ϕ ∈ D. Then C Φ is bounded on A p µ if and only ϕ converges uniformly in C ε for every ε > 0 and ϕ(C + ) ⊂ C + and in this case C Φ is a contraction.

Remark. The condition Φ(s) = c 0 s + ϕ(s) where c 0 is a nonnegative integer is an arithmetical condition: Φ has this form if and only if P • Φ ∈ D for every P ∈ P (see [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF],Th.A).

Proof.

Assume that ϕ converges uniformly on C ε for every ε > 0 and that ϕ(C + ) ⊂ C + . Let P be a Dirichlet polynomial, P • Φ is then a Dirichlet series which is bounded on C + (because Φ(C + ) ⊂ C + ) and so P • Φ belongs to H ∞ and consequently to H p . By [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF],Th.6(ii) we know that

P • Φ p A p µ = +∞ 0 (P • Φ) σ p H p dµ(σ).
Now, the key point is to use the boundedness of the composition operators on H p . We remark that for each σ > 0,

(P • Φ) σ (it) = P (Φ(σ + it)) = P σ (Φ(σ + it) -σ).
Let Φ dened on C + by Φ(s) = Φ(σ + s) -σ. We claim that Φ veries the condition of the boundedness of C Φ on H p : it can be written Φ(s) = c 0 s + ϕ(s) where ϕ converges uniformly on C ε for every ε > 0. Then it suces to check that ϕ(C + ) ⊂ C + but ϕ(s) = ϕ(s + σ) + (c 0 -1)σ for every s ∈ C + and the conclusion is clear because c 0 ≥ 1 and ϕ(C + ) ⊂ C + .

Now we apply the result on the boundedness on H p . Since c 0 ≥ 1, C Φ is a contraction on H p and we obtain (σ is xed):

(P • Φ) σ p H p = P σ ( Φ) p H p ≤ P σ p H p .
This is true for every σ > 0, so +∞ 0

(P • Φ) σ p H p dµ(σ) ≤ +∞ 0 P σ p H p dµ(σ). Hence P • Φ A p µ ≤ P A p µ . Now, by density of P in A p µ , C Φ : P ∩ A p µ → A p µ
extends to a bounded operator T : A p µ → A p µ such that T (P ) = P • Φ for every P ∈ P. Let f ∈ A p µ , there exists a sequence (P n ) ⊂ P such that (P n ) converges to f in norm, then by continuity of the point evaluation at s ∈ C 1/2 on A p µ (see [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF],Th.1):

|T (f )(s) -T (P n )(s)| ≤ δ s (A p µ ) * T (f ) -T (P n ) A p µ ≤ δ s (A p µ ) * T f -P n A p µ
and then T (P n )(s) = P n (Φ(s)) converges to T (f )(s) when n goes to innity. But by evaluation at

Φ(s) ∈ C 1/2 , P n (Φ(s)) converges to f (Φ(s)), so T (f )(s) = f (Φ(s)) = C Φ (f )(s)
and the result is proved. Now assume that Φ induces a bounded composition operator on A p µ . The result follows from ideas similar to the ones of the proof of Th.B from [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF]. We only give a sketch of the proof.

Let f ∈ A p µ . For every χ ∈ T ∞ , we can show that (see [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], Prop4.3) for every

s ∈ C 1/2 , (f • Φ) χ (s) = f χ c 0 • Φ χ (s). (∆) Since f and f • Φ belong to A p µ , (f • Φ) χ and f χ c 0 extend analytically on C + almost-surely relatively to χ . As in Prop5.1 from [9], we can show that Φ χ extends analytically on C + for every χ ∈ T ∞ . Now, let χ ∈ T ∞ such that f χ c 0 , Φ χ and (f • Φ) χ extend on C + . If Φ χ does not map C + to C + , then there exists s 0 ∈ C + such that Φ χ (s 0
) is in the imaginary axis and with an argument of connectedness, we can choose it such that Φ χ (s 0 ) = 0. Then, by (∆), f χ c 0 has an analytic extension on a small segment on the imaginary axis near Φ χ (s 0 ). We denote (p n ) n≥1 the sequence of prime numbers. In [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] it is shown that

f = n≥1 e pn √ p n log(p n ) ∈ H p
and that almost surely relatively to χ, f χ c 0 does not extend analytically to any region larger than

C + . But f ∈ A p µ because H p ⊂ A p µ .
Consequently Φ χ has to map C + to C + almost surely relatively to χ and then by Proposition 4.1 from [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], Φ(C + ) ⊂ C + . Finally, it suces to apply Th3.1 from [START_REF] Queélec | Approximation numbers of composition operators on the H 2 space of Dirichlet series[END_REF] to obtain that ϕ(C + ) ⊂ C + and ϕ converges uniformly on C ε for all ε > 0.

Remark. In the previous proof we used that Φ u -u :

C + → C + when c 0 ≥ 1.
We can see this result as an equivalent of the Schwarz's lemma in this framework.

When c 0 = 0, we obtain the following result.

Theorem 2. Let p ≥ 1 and Φ :

C 1 2 → C 1 2 be an analytic function belonging to D. Then (i) If C Φ is bounded on A p µ then Φ converges uniformly on C ε for every ε > 0 and Φ(C + ) ⊂ C 1/2 . (ii) If Φ converges uniformly on C ε for every ε > 0 and Φ(C + ) ⊂ C 1/2+η (where η > 0) then C Φ is Hilbert-Schmidt on A 2 µ , a fortiori bounded. (iii) If C Φ is bounded on A 2 µ , then it is bounded on A 2k µ for any k ≥ 1. Conse- quently C Φ is bounded on A 2k
µ for any k ≥ 1 under conditions of (ii). Proof.

(i) As in the case c 0 ≥ 1, the proof follows from [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], Th.B and some easy adaptations.

(ii) By Lemma 5 from [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] we have:

n -Φσ 2 H 2 = T ∞ 1 0 |n -Φχ(σ+it) | 2 dtdm(χ) ∀n ≥ 1.
Now Φ : C + → C 1/2+η and then by Proposition 4.1 from [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF], the same holds for Φ χ for every χ ∈ T ∞ . Consequently

n -Φσ 2 H 2 ≤ n -1-2η ∀n ≥ 1.
Now we show that C Φ is Hilbert-Schmidt and then obviously bounded. The sequence (e µ n ) n≥1 is an orthonormal basis of A 2 µ and C Φ (e µ n ) ∈ H ∞ for every n ≥ 1: indeed e µ n ∈ P and Φ(C + ) ⊂ C + . Then by [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF], Th.6(ii):

+∞ n=1 C Φ (e µ n ) 2 A 2 µ = +∞ n=1 +∞ 0 (C Φ (e µ n )) σ 2 H 2 dµ(σ) = +∞ n=1 +∞ 0 n -Φσ 2 H 2 w h (n) dµ(σ) ≤ +∞ n=1 +∞ 0 n -1-2η w h (n) dµ(σ) = +∞ n=1 n -1-2η w h (n) .
Now in [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF], it is shown that the weight (w h (n)) decreases more slowly than any negative power of n so there exists

C > 0 such that w h (n) ≥ Cn -η for every n ≥ 1, then +∞ n=1 C Φ (e µ n ) 2 A 2 µ ≤ 1 C +∞ n=1 1 n 1+η < +∞
and so C Φ is Hilbert-Schmidt.

(iii) Assume that C Φ is bounded on A 2 µ . Let P be a Dirichlet polynomial, we get

P • Φ A 2k µ = P k • Φ 1/k A 2 µ ≤ C Φ 1/k P k 1/k A 2 µ = C Φ 1/k P A 2k µ .
By density of the polynomials and boundedness of the point evaluation we obtain the result (as in the proof of Theorem 1).

Questions.

Is it true that C Φ is actually nuclear? Can we choose η = 0 in (ii) to get boundedness?

(f n ) in X which converges to 0 uniformly on C 1/2+ε for every ε > 0, one has C Φ (f n ) X → 0.
Proof. The point-evaluation at s ∈ C 1/2 is bounded on X, X has the Fatouproperty (in the sense of [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF], Prop 4.8) and there exists a Montel's theorem for H ∞ (see [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]). Then it suces to apply the same reasoning than in the proof of Proposition 4.8 from [START_REF] Lefèvre | Composition operators on Hardy-Orlicz spaces[END_REF].

Remark. With this criterion, it is easy to see that if

C Φ is compact on H 2 (resp. A 2 µ ) then C Φ is compact on H 2k (resp. A 2k µ ) for any k ≥ 1.

A Sucient conditions of compactness

In this section, we follow ideas from [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]: we dene the generalized Nevanlinna counting function associated to the spaces A 2 µ in order to study the compactness of composition operators on A 2 µ . We need some preliminaries from [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF].

For σ > 0, we dene

β h (σ) := σ 0 (σ -u)h(u) du = σ 0 t 0 h(u) dudt.
We point out that if h is continuous, the two rst derivatives of β h are

β h (σ) = σ 0 h(u)du, β h (σ) = h(σ).
Remark. If we denote β α instead of β h for the spaces A 2 α , it is easy to see that β α (σ) ≈ σ α+2 when σ is close to 0.

Theorem ("Littlewood-Paley formula" [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]). Let η be a Borel probability measure on R. Then

f 2 A 2 µ = |f (+∞)| 2 + 4 T ∞ +∞ 0 R β h (σ)|f χ (σ + it)| 2 dη(t)dσdm(χ).
Remark. For a Dirichlet series f with expansion (1), f (+∞) stands for a 1 , the constant coecient.

Notation. We shall call a c 0 -symbol any analytic function Φ :

C 1/2 → C 1/2 of the form Φ(s) = c 0 s + ϕ(s)
where c 0 is a nonnegative integer and ϕ ∈ D such that C Φ is bounded on A 2 µ . When c 0 ≥ 1 it is equivalent to the fact that ϕ converges uniformly on C ε for every ε > 0 and ϕ(C + ) ⊂ C + . We point out that in the sequel, the hypothesis "c 0 ≥ 1" will be crucial. Denition 1. We dene the generalized Nevanlinna counting function associated to a function β : (0, +∞) → (0, +∞) by

N β,Φ (s) =      a∈C+ Φ(a)=s β( (a)) if s ∈ Φ(C + ), 0 else. When β(σ) = σ, we just denote N Φ this function.
Remarks.

(i) N Φ is associated to the space H 2 , it was already dened in [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF].

(ii) N β,Φ is the equivalent in our framework of Dirichlet series of the generalized Nevanlinna counting function dened in [START_REF] Kellay | Compact composition operators on weighted Hilbert spaces of analytic functions[END_REF] in the framework of analytic functions on the unit disk.

Lemma 1. Let Φ be a c 0 -symbol with c 0 ≥ 1. For every s ∈ C + , we have

N β h ,Φ (s) = (s) 0 N Φu (s)h(u)du
where Φ u is dened by Φ u (s) := Φ(s + u). Proposition 2. Let Φ be a c 0 -symbol with c 0 ≥ 1. For every s ∈ C + , we have

Proof. Let

s ∈ C + . N β h ,Φ (s)
N β h ,Φ (s) ≤ β h ( (s)) c 0 .
Proof. By the previous lemma, we know that

N β h ,Φ (s) = (s) 0 N Φu (s)h(u)du.
We already pointed out that Φ u -u : C + → C + and then by Littlewood's inequality ( [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF],Prop.3):

N Φu-u (s ) ≤ (s ) c 0 ∀s ∈ C + .
But N Φu (s) = N Φu-u (s -u) when (s) > u > 0. Finally

N β h ,Φ (s) ≤ (s) 0 ( (s) -u)h(u) c 0 du = β h ( (s)) c 0 .
In order to obtain an upper bound for the essential norm of C Φ , we need to obtain some estimates on N β h ,Φ . We follow ideas from [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]. Proposition 3. Let Φ : C + → C + and Φ k : C + → C + where k ≥ 0, some analytic functions. Assume that (Φ k ) k≥0 converges uniformly to Φ on every compact subset of C + , then for every s ∈ C + ,

N β h ,Φ (s) ≤ lim inf k→+∞ N β h ,Φ k (s). Proof. Let us x s ∈ Φ(C + ) (if s / ∈ Φ(C + )
there is nothing to prove) and ε > 0. Let a 1 , . . . , a n ∈ C + satisfying Φ(a i ) = s for every i = 1, . . . , n and δ > 0 such that 2nδ < ε. (Φ k ) k≥0 converges uniformly to Φ on every compact set of C + and so particularly on each disk D(a i , δ) of center a i and radius δ. Then by the Hurwitz's lemma, there exists K ≥ 0 such that for any k ≥ K, s ∈ Φ k (D(a i , δ)) for every i = 1, . . . , n.

Now let us x

k ≥ K, there exist a k 1 ∈ D(a 1 , δ), ... , a k n ∈ D(a n , δ) such that Φ(a k i ) = s for every i = 1, . . . , n.
By denition of β h , we have

β h ( (a k i )) = (a k i ) 0 ( (a k i ) -u)h(u)du ≥ (ai)-δ 0 ( (a i ) -δ -u)h(u)du = β h ( (a i )) -δ (ai)-δ 0 h(u)du + (ai) (ai)-δ ( (a i ) -u)h(u)du ≥ β h ( (a i )) -2δ because h 1 = 1.
Summing up all the terms

β h ( (a k 1 )) + • • • + β h ( (a k n )) ≥ β( (a 1 )) + • • • + β( (a n )) -2nδ.
Then for every k ≥ K, we get

β( (a 1 )) + • • • + β( (a n )) ≤ 2nδ + β h ( (a k 1 )) + • • • + β h ( (a k n )) ≤ ε + N Φ k (s). So β( (a 1 )) + • • • + β( (a n )) ≤ ε + lim inf k→+∞ N Φ k (s)
and since n and ε are arbitrary, the result is proved.

Corollary 1. Let Φ a c 0 -symbol with c 0 ≥ 1. Assume that sup (s)>0 N β,Φ (s) β( (s)) = C < +∞, then sup χ∈T ∞ sup (s)>0 N β,Φχ (s) β( (s)) = C.
Proof. For every τ ∈ R, it is easy to see that N β h ,Φτ (s) = N β h ,Φ (s + ic 0 τ ) and so for every τ ∈ R we have

sup (s)>0 N β,Φτ (s) β( (s)) = C.
Now for χ ∈ T ∞ , Φ χ is a vertical limit of a sequence (Φ τn ) n≥0 where (τ n ) n≥0 ⊂ R and then it suces to apply Proposition 3 to obtain that

sup (s)>0 N β,Φχ (s) β( (s)) ≤ C
The other inequality is true because Φ = Φ χ with χ = (1, 1, . . . ).

Remark. In the previous corollary, we can take the supremum on a band of the form {s ∈ C + , (s) < θ} where θ ∈ R.

Let H a Hilbert space and K(H) the space of all compact operators on H. Let T be a bounded operator on H, we recall that its essential norm is dened by

T H,e = inf{ T -K , K ∈ K(H)}.
Clearly, the essential norm of T is zero if and only if T ∈ K(H). The next result will be very useful.

Proposition. ([20],Prop5.1) Let T be a bounded operator on H and (K n ) n≥1 a sequence of self-adjoint compact operators on H. Let R n = I -K n . Assume that R n = 1 for every n ≥ 1 and that (R n ) n≥1 converges pointwise to zero. Then

T H,e = lim n→+∞ T R n . Theorem 3. Let Φ a c 0 -symbol with c 0 ≥ 1. Assume that Im(ϕ) is bounded on C + , then C ϕ A 2 µ ,e ≤ 2 sup C+ |Im(ϕ)| + c 0 • lim sup (s)→0 N β h ,Φ (s) β h (Re(s))
and

C ϕ H 2 ,e ≤ 2 sup C+ |Im(ϕ)| + c 0 • lim sup (s)→0 N Φ (s) (s) .
Corollary 2. Assume that Im(ϕ) is bounded on C + then:

(i) If N β h ,Φ (s) = o(β( (s)) when (s) → 0 then C Φ is compact on A 2 µ . (ii) If N Φ (s) = o( (s)) when (s) → 0 then C Φ is compact on H 2 .
Remark. Cor2.(ii) is already known (see [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF], Th2). Proof of Th.3. We only give the proof in the case of A 2 µ , the proof in the case H 2 is essentially the same.

For n ≥ 1 and f ∈ A 2 µ which has form (1) we dene

           K n (f ) = n k=1 a k e k R n (f ) = (I -K n )(f ) = +∞ k=n+1 a k e k .
Each K n is a self-adjoint compact operator on A 2 µ with norm 1 because K n is a projection on a nite linear subspace and (R n ) clearly converges pointwise to zero. Then by the previous proposition

C Φ A 2 µ ,e = lim n→+∞ C Φ R n = lim n→+∞ sup f A 2 µ ≤1 C Φ (R n (f )) A 2 µ .
In the sequel, we shall work on Ω = (0, +∞) × (0, 1) with the area measure: we apply the Littlewood-Paley formula for A 2 µ with the Lebesgue measure on (0, 1):

C Φ (R n (f )) 2 A 2 µ = |R n (f ) • Φ(+∞)| 2 + 4 T ∞ Ω β h ( (s))|(R n (f ) • Φ) χ (s)| 2 dsdm(χ).
We can point out that R n (f ) • Φ(+∞) = 0 for every n ≥ 2 because c 0 ≥ 1 and then Φ(+∞) = +∞. Now by Proposition 4.3 from [START_REF] Gordon | The composition operators on the space of Dirichlet series with square summable coecients[END_REF]:

C Φ (R n (f )) 2 A 2 µ = 4 T ∞ Ω β h ( (s))|R n (f ) χ c 0 (Φ χ (s))| 2 |Φ χ (s)| 2 dsdm(χ).
Now we make the following "change of variables": w = Φ χ (s), Φ χ is not necessarily injective but we use the generalized formula of change of variable which involves the generalized Nevanlinna counting function (see [START_REF] Federer | Geometric Measure Theory[END_REF] for example). We get:

Ω β h ( (s))|R n (f ) χ c 0 (Φ χ (s))| 2 |Φ χ (s)| 2 ds = Φχ(Ω) |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) ds.
Im(ϕ) is bounded then there exists A > 0 such that |Im(ϕ)| < A and the same inequality holds for ϕ χ for every χ ∈ T ∞ instead of ϕ: indeed ϕ χ is a vertical limit of ϕ. Now, when s ∈ Ω, we have Im(s) ∈ (0, 1) and

w = Φ χ (s) veries Im(w) = c 0 Im(s) + Im(ϕ χ (s)) so Im(w) ∈ [-A, A + c 0 ]. Finally if we denote Q = (0, +∞) × (-A, A + c 0 ), we get Ω β h ( (s))|R n (f ) χ c 0 (Φ χ (s))| 2 |Φ χ (s)| 2 ds ≤ Q |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) ds.
Let us x θ > 0, we write Q = Q θ ∪ Qθ where Q θ = (0, θ) × (-A, A + c 0 ) and Qθ = (θ, +∞) × (-A, A + c 0 ). We shall cut the previous integral in two parts. Let γ θ dened by

γ θ = sup χ∈T ∞ sup 0<Re(s)<θ N β h ,Φχ (s) β h ( (s)) = sup 0<Re(s)<θ N β h ,Φ (s) β h ( (s)) .
The second equality is true by Corollary 1. We have

T ∞ Q θ |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) dsdm(χ) ≤ γ θ T ∞ Q θ |R n (f ) χ c 0 (s)| 2 β h ( (s)) dsdm(χ).
Now we take the supremum:

sup f A 2 µ ≤1 T ∞ Q θ |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) dsdm(χ) ≤ γ θ sup f A 2 µ ≤1 T ∞ Q θ |R n (f ) χ c 0 (s)| 2 β h ( (s)) dsdm(χ) ≤ γ θ sup f A 2 µ ≤1 T ∞ Q θ |f χ c 0 (s)| 2 β h ( (s)) dsdm(χ) .
In the last inequality, we use the fact that if f belongs to the unit ball of A 2 µ , R n (f ) too. Finally, by using again the Littlewood-Paley formula with the normalized Lebesgue measure on (-A, A + c 0 ) and the fact that the map χ → χ c0 is measure preserving we obtain:

sup f A 2 µ ≤1 4 T ∞ Q θ |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) dsdm(χ) ≤ (2A + c 0 )γ θ .
Now we obtain an upper bound for the second integral. By Proposition 2:

T ∞ Qθ |R n (f ) χ c 0 (s)| 2 N β h ,Φχ (s) dsdm(χ) ≤ T ∞ Qθ |R n (f ) χ c 0 (s)| 2 β h ( (s)) c 0 dsdm(χ) (∆)
Recall (see [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF], lem2) that for every f ∈ H 2 with form (1) and every probability measure η on R we have for every σ > 0:

T ∞ R |f χ (σ + it)| 2 dη(t)dm(χ) = +∞ k=2 |a k | 2 k -2σ log 2 (k), ∀σ > 0.
Hence, we get:

(∆) = 2A + c 0 c 0 +∞ θ +∞ k=n+1 β h (σ)|a k | 2 k -2σ log 2 (k)dσ.
Let us x ε > 0. We claim that there exists K = K θ > 0 such that for every

k ≥ K: +∞ θ β h (σ)k -2σ log 2 (k)dσ ≤ εw h (k).
Indeed, we know by [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF] that (w h (k)) is a decreasing sequence that decays more slowly than any negative power, so there exists C = C θ > 0 such that:

w h (k) ≥ Ck -θ ∀k ≥ 1. Then +∞ θ β h (σ) k -2σ log 2 (k) w h (k) dσ ≤ +∞ θ σk θ-2σ C -1 log 2 (k)dσ because β h (σ) ≤ σ by denition of β h .
Clearly the right-hand side of the inequality goes to 0 when k goes to innity.

Finally when n ≥ K, we obtain

2A + c 0 c 0 +∞ θ +∞ k=n+1 β h (σ)|a k | 2 k -2σ log 2 (k)dσ ≤ 2A + c 0 c 0 × ε
because we work with functions in A 2 µ with norm less than 1. Now summing up the two integrals, we obtain that for n ≥ K,

C Φ R n ≤ (2A + c 0 )γ θ + 2A + c 0 c 0 × ε ∀n ≥ K.
At last,

C Φ A 2 µ ,e = lim n→+∞ C Φ R n ≤ (2A + c 0 )γ θ + 2A + c 0 c 0 × ε.
Now since ε and θ are arbitrary, we obtain the result.

B Necessary conditions of compactness

First, we recall some facts from [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF]. The reproducing kernel at

s ∈ C 1/2 associated to A 2 µ is dened for every w ∈ C 1/2 by K µ,s (w) := +∞ n=1 n -s-w w h (n) .
Example. Let α > -1. The reproducing kernel at s ∈ C 1/2 associated to the space A 2 α is denoted by K α,s instead of K µα,s and we have for every

w ∈ C 1/2 , K α,s (w) = +∞ n=1 (1 + log(n)) α+1 n -s-w .
We saw in the previous section that the compactness is linked with the behaviour of Φ near to the imaginary axis. Then, as in [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF], we shall work with partial reproducing kernels which are dened on C + and not only on C 1/2 . Denition. Let s ∈ C + and l ≥ 1. On A 2 µ , we dene the partial reproducing kernel of order l at s ∈ C + by

K l µ,s (w) := n≥1 p + (n)≤p l n -s-w w h (n)
where p + (n) is the greatest prime divisor of n.

We claim that these partial reproducing kernels are dened on C + : indeed we know by [START_REF] Mccarthy | Hilbert spaces of Dirichlet series and their multipliers[END_REF] that the weight (w h (n)) decreases more slowly than any negative power of n so if ε > 0, there exists C > 0 such that w h (n) > Cn -ε for every n ≥ 1. Then

n≥1 p + (n)≤l n -(s)-(w) w h (n) ≤ 1 C n≥1 p + (n)≤l 1 n (s)+ (w)-ε = 1 C l i=1 1-p -(s)-(w)+ε i -1 .
For s ∈ C + , this quantity is then bounded for any w ∈ C ε and any ε > 0 and so the claim is proved.

We shall need the following proposition which follows easily from [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF],Prop.5.

Proposition 4. Let Φ a c 0 -symbol with c 0 ≥ 1 and l ≥ 1. Assume that Φ(s) =

c 0 s + +∞ n=1 c n n -s where c n = 0 when p + (n) > l. Then C * Φ (K l µ,s ) = K l µ,Φ(s) .
Lemma 2. On the unit ball B A 2 µ of A 2 µ , the weak topology is the topology of uniform convergence on every half-plane C 1/2+ε with ε > 0.

Proof. Let s ∈ C + , we have

N β h ,Φ (s) = a∈C+ Φ(a)=s β h ( (a)) = a∈C+ Φ(a)=s G( (a)) (a).
Let ε > 0, we know that

lim (s)→0 (s) (Φ(s)) = 0
and then by Condition (κ), there exists δ > 0 such that

G( (s)) ≤ εG( (Φ(s)))
when (s) < δ. Now, thanks to the Schwarz's lemma if Φ(a) = s and (s) < δ then (a) < δ too. Consequently

N β h ,Φ (s) = a∈C+ Φ(a)=s G( (a)) (a) ≤ ε a∈C+ Φ(a)=s G( (Φ(a))) (a) = εG( (s)) a∈C+ Φ(a)=s (a)
= εG( (s))N Φ (s)

for every s ∈ C + such that (s) < δ. Now by Littlewood's inequality we get: c n n -s where c n = 0 when p + (n) > l and that Im(ϕ) is bounded, then the following assertions are equivalent:

N β h ,Φ (
(i) C Φ is compact on A 2 α .
(ii) lim Remark. We point out that the condition (ii) does not depend on α.

Denition. Let Φ : C + → C + be an analytic function and k ∈ N, we say that Φ is k-valent if for every w ∈ C + , there exists at most k solutions to Φ(s) = w.

If Φ is k-valent for some k ∈ N then we shall say that Φ is nite-valent.

Theorem 6. Let l ≥ 1 and Φ be a c 0 -symbol with c 0 ≥ 1 such that Φ(s) =

c 0 s + +∞ n=1 c n n -s where c n = 0 if p + (n) > l.
Assume that Im(ϕ) is bounded and that Φ is nite-valent, then the following assertions are equivalent:

(i) C Φ is compact on H 2 . (ii) lim (s)→0 (s) (Φ(s)) = 0.
(iii) N Φ (s) = o( (s)) when (s) goes to 0.

Proof. We only have to proof that (ii) ⇒ (iii): the other implications have been proven in [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]. We assume that Φ is k-valent for some k ≥ 0. Let ε > 0 and s ∈ C + , as in the proof of Proposition 5 we can show that there exists δ > 0 such that for every ∈ C + with (s) < δ:

N Φ (s) ≤ ε (s) a∈C+ Φ(a)=s 1 ≤ εk (s). Remark. Let l ≥ 1 and Φ a c 0 -symbol with c 0 ≥ 1 such that Φ(s) = c 0 s + +∞ n=1 c n n -s where c n = 0 if p + (n) > l. If Im(ϕ) is bounded and C ϕ is compact on H 2 then lim (s)→0 (s) (Φ(s)) = 0
and consequently C Φ is compact on A 2 α for any α > -1.

Questions. Is the previous implication true for general c 0 -symbols?

Compactness and Carleson measures

In this section, we are going to obtain a sucient condition of compactness for C Φ with a "Carleson-measure" condition. We still work on A 2 µ where µ is a probability measure on (0, +∞) such that 0 ∈ Supp(µ).

When Φ is a c 0 -symbol with c 0 ≥ 1, it is shown in [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF] that Φ admits radial limits and so if λ is the Lebesgue measure on R, Φ * (it) := lim σ→0 + Φ(σ + it) exists λ-almost everywhere.

Notation. Let t ∈ R and h > 0. We dene the Carleson window centered at it and of size h by H(t, h) := {s ∈ C + , |s -it| < h}. (ii) Let λ µ = λ ⊗ µ. We denote by λ µ,Φ the pullback measure of λ µ by Φ, ie, for every open set Ω ⊂ C + :

λ µ,Φ (Ω) := λ µ ({s ∈ C + , Φ(s) ∈ Ω}).
The Carleson function associated to λ Φ is then dened by

ρ µ,Φ (h) := sup t∈R λ µ,Φ (H(t, h)).
In the sequel we shall denote β instead of β h . The next theorem is the main result of this section. Theorem 7. Let Φ be a c 0 -symbol with c 0 ≥ 1. There exists K > 0 (independent of Φ) such that:

(i) sup s∈H(t,h/2) N Φ (s) ≤ K • λ Φ H(t, 2c 0 h)
for every h small enough.

In particular:

sup s∈C+ (s)<h/2 N Φ (s) ≤ K • ρ Φ 2c 0 h for every h small enough. (ii) sup s∈H(t,h/2) N β,Φ (s) ≤ K • λ µ,Φ H(t, 2c 0 h)
for every h small enough.

In particular:

sup s∈C+ (s)<h/2 N β,Φ (s) ≤ K • ρ µ,Φ 2c 0 h
for every h small enough. Corollary 4. Let Φ be a c 0 -symbol with c 0 ≥ 1 such that Im(ϕ) is bounded on C + . Then there exists C > 0 such that:

(i)

C Φ H 2 ,e ≤ C lim sup h→0 ρ Φ (h) h .

In particular if ρ

Φ (h) = o(h) when h → 0 then C Φ is compact on H 2 . (ii) C Φ A 2 µ ,e ≤ C lim sup h→0 ρ µ,Φ (h) β(h) . In particular if ρ µ,Φ (h) = o(β(h)) when h → 0 then C Φ is compact on A 2 µ .
Example. When µ = µ α , we denote N α (resp. ρ α,Φ ) the corresponding generalized Nevanlinna counting function (resp. the Carleson function). We know that in that case, β α (σ) ≈ σ α+2 when σ is small and then if Im(ϕ) is bounded:

C Φ e,A 2 α ≤ C lim sup h→0 ρ α,Φ (h) h α+2 .
Proof of Cor.4. We only give the proof of (i). Let s ∈ C + with (s) small enough, we have

N Φ (s) ≤ sup s ∈H(Im(s),2 (s)) N Φ (s ) ≤ K • ρ Φ 8c 0 (s)
by theorem 7. Then we obtain

N Φ (s) (s) ≤ 8c 0 K • ρ Φ 8c 0 (s) 8c 0 (s)
and so the result.

In order to prove Theorem 7, we need to recall some facts about Carleson measures and the classical Nevanlinna counting function on the unit disk.

Let φ : D → D be an analytic self-map. The Nevanlinna counting function of φ is dened for w ∈ φ(D) by

N φ (w) = z∈D φ(z)=w log(1/|z|).

When w /

∈ φ(D), N φ (w) = 0 by convention. We denote by m φ the pullback measure of the Lebesgue measure on the unit circle by φ * , the boundary values function of φ (which exists because φ ∈ H ∞ (D)). By abusing the notation, we still denote φ instead of φ * .

For η ∈ T and h > 0, the Carleson window S(η, h) is dened by

S(η, h) := {z ∈ D, |z -η| < h}.
We shall make a crucial use of the following result (see [START_REF] Lefèvre | Nevanlinna counting function and Carleson function of analytic maps[END_REF],Th1.1): Let φ : D → D an analytic self-map. For every β > 1, there exists a universal constant C β > 0 such that:

(1/C β )m φ (S(ξ, h)) ≤ sup z∈S(ξ,h) N φ (z) ≤ C β m φ (S(ξ, βh))
for every 0 < h < (1 -|φ(0)|)/β and every ξ ∈ T.

Proof of Th.7(i).

Step one: We prove the result when t = 0. We shall use a family of functions (ψ ξ ) ξ>0 where ψ ξ is dened by

   ψ ξ : C + -→ D s -→ s -ξ s + ξ .

Claim. For every

h > 0, H(0, h/2) ⊂ ψ -1 c0ξ (S(-1, h/ξ)).
Let us prove this fact:

s ∈ ψ -1 c0ξ (S(-1, h/ξ)) ⇔ ψ c0ξ (s) ∈ S(-1, h/ξ) ⇔ s -c 0 ξ s + c 0 ξ + 1 < h ξ ⇔ 2s s + c 0 ξ < h ξ ⇔ |s| < h 2 s + c 0 ξ ξ .
Now if s ∈ H(0, h/2) then |s| < h 2 and it suces to remark that for every ξ > 0,

|s + c 0 ξ| ξ ≥ (s + c 0 ξ) ξ ≥ c 0 ≥ 1,
and this justies the claim.

Let h > 0 and s ∈ H(0, h/2). Then w ξ := ψ c0ξ (s) ∈ S(-1, h/ξ). Assume that s ∈ Φ(C + ) (if it is not the case, there is nothing to prove) and let s 1 , . . . , s n ∈ C + such that Φ(s i ) = s for every i = 1, . . . , n.

For ξ > 0, we dene the analytic self-map Θ ξ : D → D by Θ ξ = ψ c0ξ •Φ•ψ -1 ξ . We note that Θ ξ (ψ ξ (s i )) = ψ c0ξ (Φ(s i )) = ψ c0ξ (s) = w ξ and so For i ∈ {1, . . . , n} we have

log(1/|ψ ξ (s i )|) = 1 2 log |s i | 2 + 2ξ (s i ) + ξ 2 |s i | 2 -2ξ (s i ) + ξ 2 = 1 2 log 1 + 4ξ (s i ) |s i | 2 -2ξ (s i ) + ξ 2 .
Let ε > 0. For ξ large enough we obtain that for every i ∈ {1, . . . , n}:

2(1 -ε) (s i ) ξ ≤ log(1/|ψ ξ (s i )|).
And so

2(1 -ε) n i=1 (s i ) ξ ≤ N Θ ξ (w ξ ).
Now using with β = 2 and remembering that w ξ ∈ S(-1, h/ξ) for every ξ > 0, we obtain

N Θ ξ (w ξ ) ≤ C 2 m Θ ξ (S(-1, 2h/ξ))
for every 0 < h/ξ < (1 -|Θ ξ (0)|)/2. Point out that Θ ξ (0) = ψ c0ξ (Φ(ξ)) = ϕ(ξ) 2c 0 ξ + ϕ(ξ) and this quantity goes to 0 when ξ goes to innity. So we dene

m 0 = inf ξ>1 ξ(1 -|Θ ξ (0)|)/2 > 0.
It is clear that m 0 ≤ 1/2. Let us point out that m 0 = m 0 (ϕ) = m 0 (ϕ(• + iτ )) for any τ ∈ R (this remark will be useful in step 2). When h < m 0 , we get Then for ξ > 1 large enough we obtain (1-h/ξ) < c0h

(1-h) < 2c 0 h since h < m 0 ≤ 1/2. Then

2(1 -ε) n i=1 Re(s i ) ≤ C 2 m Φ (H(0, 2c 0 h)).
Since n is arbitrary we get:

2(1 -ε)N Φ (s) ≤ C 2 m Φ (H(0, 2c 0 h)).
for any ε > 0 and the rst step is proved.

Step two: we prove now the result when t = 0. We have Applying step one to Φ, we get for every h small enough. Now it suces to remark that

  ) -u)h(u)du thanks to the "Schwarz's lemma": indeed Φ(C (a) ) ⊂ C (a) and then (a) > (s) implies Φ(a) = s. Now by the Fubini's theorem:N β h ,Φ (s) =(s) 0 u< (a)≤ (s) Φ(a)=s ( (a) -u)h(u) du. Now it suces to remark that u< (a)≤ (s) Φ(a)=s ( (a) -u) = u< (a) Φu(a-u)=s ( (a) -u)

  s) ≤ εG( (s)) (s) c 0 = εβ h ( (s)) c 0 when (s) < δ and this proves the result. Theorem 5. Let α > -1, Φ a c 0 -symbol with c 0 ≥ 1 and l ≥ 1. Assume that Φ(s) = c 0 s + +∞ n=1

  N α,Φ (s) = o(( (s)) α+2 ) when (s) goes to 0.Proof. It suces to use Corollary 3, Proposition 5 and Corollary 2.

  (i) We denote λ Φ the pullback measure of λ by Φ * , ie, for every open setΩ ⊂ C + : λ Φ (Ω) := λ({t ∈ R, Φ * (it) ∈ Ω}).The Carleson function associated to λ Φ is then dened by ρ Φ (h) := sup t∈R λ Φ (H(t, h)).

  |ψ ξ (s i )|) ≤ N Θ ξ (w ξ ).

N

  Θ ξ (w ξ ) ≤ C 2 m Θ ξ (S(-1, 2h/ξ)) and m Θ ξ (S(-1, 2h/ξ)) = m({η ∈ T, Θ ξ (η) ∈ S(-1, 2h/ξ)}) = m({η ∈ T, Ψ c0ξ • Φ * • ψ -1 ξ (η) ∈ S(-1, 2h/ξ)}) = {Ψ c 0 ξ •Φ * •ψ -1 ξ (e iθ )∈S(-1,2h/ξ)}) dθ 2π .Now we make the following change of variables iu = ψ -1ξ (e iθ ):m Θ ξ (S(-1, 2h/ξ)) = {Φ * (iu)∈Ψ -1 c 0 ξ (S(-1,2h/ξ))} ξ π(ξ 2 + u 2 ) du = {|Φ * (iu)|<h |Φ * (iu)+c 0 ξ| ξ } ξ π(ξ 2 + u 2 )du.

N

  Φ (s + ic 0 t).But ifΦ(s) := c 0 s + ϕ(s + it) = Φ(s + it) -ic o t, we obtain for s ∈ H(0, h/2): N Φ (s + ic 0 t) = a∈C+ Φ(a)=s+ic0t (a) = a∈C+ c0(a-it)+ϕ(it+a-it)=s (a -it) = a ∈C+ Φ(a )=s (a )= N Φ(s).

N

  Φ (s + ic 0 t) = sup s∈H(0,h/2) N Φ(s) ≤ Kλ Φ(H (0, 2c 0 h))

λ

  Φ(H (0, 2c 0 h)) = λ({t ∈ R, Φ * (it ) ∈ H(0, 2c 0 h)}) = λ({t ∈ R, |Φ * (it + it ) -ic 0 t| < 2c 0 h}) = λ({t ∈ R, |Φ * (it ) -ic 0 t| < 2c 0 h}) = λ Φ (H(c 0 t, 2c 0 h)).

4

  Compactness and Nevanlinna counting functionLet p ≥ 1 and X = H p or A p µ . We begin this section with the following criterion of compactness: Proposition 1. Let Φ : C 1/2 → C 1/2 be an analytic function which induces a bounded composition operator on X. Then C Φ is compact on X if and only if for every bounded sequence

→ C 1 2 such that for any f ∈ H 2 , f • Φ ∈ H 2 .We denote by D the class of functions f which admit representation by a convergent Dirichlet series in some half-plane and for θ ∈ R, C θ will be the following half-plane {s ∈ C, (s) > θ}. We shall denote C + instead of C 0 .

Acknowledgements. I would like to thank Pascal Lefèvre for his advice regarding this paper.

Proof. Let (f n ) ⊂ B A 2 µ which converges weakly to 0. Let ε > 0, by the result on the point evaluation from [START_REF] Bailleul | Some Banach spaces of Dirichlet series[END_REF], there exists a constant C ε such that

and then the sequence (f n ) is uniformly bounded on C 1/2+ε . Now by the Montel's theorem for H ∞ (see [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF],Lem18), there exists a subsequence which converges uniformly on C 1/2+ε . It suces to show now that 0 is the unique cluster point, but the point evaluation is bounded and (f n ) converges weakly to 0 and so 0 is the unique cluster point.

Conversely let

µ which converges uniformly to 0 on C 1/2+ε for every ε > 0. There exists at least a subsequence which converges weakly to

But the point evaluation is bounded for every s ∈ C 1/2 and (f n ) converges uniformly to 0 on C 1/2+ε for every ε > 0 so f = 0. Theorem 4. Let Φ a c 0 -symbol with c 0 ≥ 1 and l ≥ 1. Assume that Φ(s) = c 0 s + +∞ n=1 c n n -s where c n = 0 when p + (n) > l. We have:

Proof. Let l ≥ 1. For s ∈ C + , we dene the following family of functions:

This family is contained in B A 2 µ and converges uniformly to 0 on C 1/2+ε for every ε > 0: indeed we have

and for every w ∈ C 1/2+ε :

because the partial reproducing kernels are dened on

as well and by the previous lemma we know that (k s ) converges weakly to 0 so K * (k s ) is norm convergent to 0. Then

and then

Now with Proposition 4, and the fact that the inequality is true for every K ∈ K(A 2 µ ), we obtain the result.

Corollary 3. Let Φ a c 0 -symbol with c 0 ≥ 1 and l ≥ 1. Assume that Φ(s) =

c n n -s where c n = 0 when p + (n) > l. Assume that C Φ ∈ K(A 2 µ ), then:

In particular if

Remark. The result concerning A 2 α is already known (see [START_REF] Bayart | Compact composition operators on a Hilbert space of Dirichlet series[END_REF]).

C A criterion of compactness

In this section, we still work on A 2 µ where µ is a probability measure such that dµ(σ) = hdσ with h a positive continuous function. Recall that

In the spirit of [START_REF] Kellay | Compact composition operators on weighted Hilbert spaces of analytic functions[END_REF] (see p12) we introduce a new condition: we say that a weight

Examples.

(i) When h is a non decreasing function then (κ) is fulllled for β h : indeed in this case we extend continuously G at 0 by G(0) = 0 and it suces to check that G is a convex function but it is easy to see that

and

Proof of Th.7(ii). By Lemma 1, we know that

Now in the proof of (i), we point out that we can choose the same m 0 for Φ u for every u ≥ 0 and so for s ∈ H(t, h/2) with h < m 0 we have thanks to the Fubini's theorem.