Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Orthogonal decomposition of composition operators on the H

space of Dirichlet series

OleFredrik Brevig^a,∗, Karl-Mikael Perfekt^b

aDepartmentofMathematics,UniversityofOslo,0851Oslo,Norway

bDepartmentofMathematicalSciences,NorwegianUniversityofScienceand Technology(NTNU),NO-7491Trondheim,Norway

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received3September2021 Accepted7December2021 Availableonline14December2021 CommunicatedbySophieGrivaux

MSC:

primary47B33

secondary30B50,30H10

Keywords:

Dirichletseries Hardyspaces

Compositionoperators

Let _H² denote the Hilbert space of Dirichlet series with square-summable coefficients.We study composition opera- tors_Cϕon _H² whicharegeneratedby symbolsoftheform ϕ(s)=c0s+

n≥1cnn^−s,inthecasethatc0≥1.Ifonlya subsetP ofprimenumbersfeaturesintheDirichletseriesof ϕ,thentheoperator_Cϕadmitsanassociatedorthogonalde- composition.UndersparsenessassumptionsonPweusethis toasymptoticallyestimatetheapproximationnumbersof_Cϕ. Furthermore,inthecasethatϕissupportedonasingleprime number,weaffirmativelysettletheproblemofdescribingthe compactnessof_CϕintermsoftheordinaryNevanlinnacount- ingfunction.Wegive detailedapplicationsofour resultsto affinesymbolsandtoanglemaps.

©2021TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

✩ K.-M.Perfektwas partiallysupportedby grant EP/S029486/1of theUKEngineering andPhysical SciencesResearchCouncil(EPSRC).

* Correspondingauthor.

E-mailaddresses:[email protected](O.F. Brevig),[email protected](K.-M. Perfekt).

https://doi.org/10.1016/j.jfa.2021.109353

0022-1236/©2021TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Let _H² be the Hilbert space of Dirichlet series f(s) =

n≥1b_nn^−s with square- summable coefficients. For real numbers θ, set Cθ = {s ∈ C : Res > θ}, and let ϕ:C_1/2→C_1/2 be ananalytic function.Gordon andHedenmalm[10] established that the compositionoperatorCϕf =f ◦ϕdefinesabounded compositionoperatoron H² ifandonlyifϕbelongstotheGordon–Hedenmalmclass_G.

Definition. TheGordon–Hedenmalmclass G consistsoftheanalyticfunctionsϕ:C_1/2→ C_1/2 oftheform

ϕ(s) =c₀s+ ∞ n=1

c_nn^−s=c₀s+ϕ₀(s),

where c₀ isanon-negativeintegerandtheDirichletseriesϕ₀convergesuniformlyinC_ε foreveryε>0 andsatisfiesthefollowingmappingproperties:

(a) Ifc0= 0,thenϕ0(C0)⊆C_1/2.

(b) Ifc₀≥1,theneitherϕ₀(C₀)⊆C₀ orϕ₀≡iτ forsomeτ∈R.

Wewill usethenotation G0and G_≥1,respectively,forthesubclasses(a)and(b).

Let T be a bounded operator on a Hilbert space. The nth approximation number an(T) isthedistanceintheoperatornormfromTtotheoperatorsofrank< n.Studying the decayof approximation numbersis relevantfor compact operators T.Indeed, T is compact ifandonlyifa_n(T)→0 asn→ ∞.

Previously, precise results for the approximation numbers of composition operators on H² haveprimarily been available for symbols ϕ∈ G0, see [4,5,15]. For case(b) of the Gordon–Hedenmalm class, the following theorem, extracted from the proofs of [5, Thm. 1.2] and[5,Thm. 8.1],givesthebestknownestimatesforgeneralϕ∈G≥1. Here, and throughoutthepaper,wedefine

ϑ= inf

s∈C0

Reϕ₀(s) (1.1)

forsymbols ϕ∈G.

Theorem 1.1(Bayart–Queffélec–Seip[5]). Supposethat ϕ∈G≥1.Then

p⁻_n^Rec1 ≤a_n(_Cϕ)≤n^−ϑ (1.2) where pn denotesthenthprimenumber.

Since the proof of Theorem 1.1 is fairly short,we will present it in ourpreliminary section.Note thattheasymptotic estimatepn ∼nlogn asn→ ∞is adirectcorollary oftheprimenumbertheorem.

Togiveanexample,supposethatϕ(s)=c0s+c1forsomec0≥1 andc1∈C0.Then ϑ= Rec₁,andife_n(s)=n^−sforn= 1,2,3,. . .denotesthestandardbasisof_H², then

Cϕen=n^−c1en^c0.

Hencean(Cϕ)=n^−Rec1 =n^−ϑ inthiscase, coincidingwith theupper boundof (1.2).

Notethatforallother symbols,where ϕ0(s)≡c1, themaximumprincipleimpliesthat ϑ<Rec₁.

One of the main goals of the present paper is to improve on the estimates (1.2) for certain symbols ϕ. Specifically, we shall place restrictions on the prime numbers appearingintheDirichletseriesϕ0.LetP denoteasetofprimenumbersandset M(P)= {n∈N : p|n =⇒ p∈P}. WesaythataDirichletseriesf issupported onP if

f(s) =

n∈M(P)

b_nn^−s.

Asetof primenumbersP iscalledsparse if

p∈Pp⁻¹<∞.Ourfirstmainresultisthe followingimprovementofthelowerbound inTheorem1.1.

Theorem1.2. Supposethatϕ∈G≥1.Ifϕ₀ issupportedonasparsesetofprimenumbers, thenforevery ε>0thereisapositiveconstant C=C(ϕ0,ε)suchthat

an(Cϕ)≥Cn^−ϑ−ε.

Ourproofof Theorem1.2 relies onanorthogonaldecompositionof Cϕ thatis made availablebytheassumptionsthatc₀≥1 andthatϕ₀issupportedonP,seeLemma3.1.

LetP^⊥denotethesetofprimenumbersnotinP.Toapplytheorthogonaldecomposition effectively,werequirethatP issparse,sothattheset M(P^⊥) haspositivedensityinN.

Wealsohaveamorerefinedresult.WesaythatasetofprimenumbersP isν-sparse forsome0< ν≤1 if

p∈Pp^−ν <∞.Inparticular, asetofprimenumbersis1-sparse ifandonlyifitissparse.

Theorem1.3. Consider asymbolϕ∈G≥1 andsuppose that ϕ0 is supportedonP.

(a) If P issparse,thenthere isaconstant C1=C1(ϕ0)suchthat a_n(_Cϕ)≥C₁Cϕe_nH².

(b) If P is ν-sparse forsome 0< ν < 1and 2ϑ≥ν/(1−ν), then there is aconstant C2=C2(ϕ0,ν)suchthat

an(Cϕ)≤C2CϕenH².

To exemplify the typeof estimates which canbe obtainedfrom Theorem 1.3, letP be aset ofprimenumbersandconsider theaffinesymbol

ϕ(s) =c₀s+c₁+

p∈P

c_pp^−s. (1.3)

Theapproximationnumbersofcompositionoperatorsgeneratedbyaffinesymbolsϕ∈G0

havebeen investigatedbyQueffélecandSeip[15,Thm. 1.3] andbyMuthukumar,Pon- nusamy,andQueffélec[13,Thm. 4.1].UsingTheorem1.3,weshallobtainthefollowing estimate for the approximation numbers of composition operators generated by affine symbols ϕ∈G_≥1.

Corollary1.4. Supposethatϕisanaffinesymbol (1.3)withc0≥1,|P|=d<∞,cp= 0 and ϑ>0.Thenϕ∈G andforn≥2,

a_n(_Cϕ)n^−ϑ(logn)^−d4.

Note that the case ϑ = 0 is omitted from Corollary 1.4. In this case the estimate from Theorem 1.3(a) failsto besharp, sinceitfollowsfrom [4,Thm. 1] that _Cϕis not compact,andthusthatan(Cϕ)1 forn≥1.InTheorem4.2weshallalsoconsidersome examplesofaffinesymbolssupportedoninfinitebutverysparsesetsofprimenumbers.

In the second part of thepaper, we will investigate when thecomposition operator Cϕ is compact on _H². Suppose thatϕ ∈ G≥1, and consider the Nevanlinna counting function

Nϕ(w) =

s∈ϕ⁻¹({w})

Res, (1.4)

definedforeveryw∈C0.Bayart[3,Prop. 3] employedtheclassicalLittlewoodinequality for the Nevanlinna counting functionin theunit discto establish the Littlewood–type estimate

Nϕ(w)≤Rew c0

. (1.5)

On account of J. Shapiro’s characterization of the compact composition operators on the Hardyspaceof theunitdisc [17],and theLittlewood–typeestimate (1.5),it seems plausiblethatthecompactnessof Cϕ on H²isrelated totherequirementthat

Relimw→0⁺

Nϕ(w)

Rew = 0. (1.6)

Bayart[3,Thm. 2] provedthatifImϕ0 isboundedand(1.6) holds,thenCϕ iscompact on H². Conversely,Bailleul [1, Thm. 6] establishedthatif ϕ0 is supportedonafinite setofprimenumbers,ϕisfinitely valent,and Cϕ iscompacton H²,then(1.6) holds.

Wegiveacompletedescriptioninthecasethatϕ₀ issupportedonasingleprime.

Theorem 1.5.Suppose that ϕ∈G_≥1 and that ϕ0 issupported on P ={p}. Then Cϕ is compacton_H² if andonlyif

Relimw→0⁺

Nϕ(w) Rew = 0.

To prove this theorem, we will exploit the fact that such functions ϕ0 are periodic (withperiod 2πi/logp),in additionto the orthogonaldecomposition discussed earlier.

Accordingly,we will decomposethe Nevanlinna countingfunction (1.4) intoaninfinite numberofrestrictedcountingfunctions.Tohandletheserestrictedcountingfunctionswe willrelyonsomeideasandtechniquesfromourrecentpaper[7],wherethecompactness of Cϕwascharacterizedinthecasethatϕ∈G0.Eachrestrictedcountingfunctioncomes with achangeof variable formula, also knownas aStanton formula, thatallows us to expressCϕfH² forDirichletseriesf ofacertainform,seeLemma6.2.

To conclude the paper we will provide a detailedstudy of angle maps. For c0 ≥ 1, ϑ≥0 and0< α <1,consider thesymbol ϕα,ϑ(s)=c0s+ϑ+ Φα(p^−s),where

Φ_α(p^−s) =

1−p^−s 1 +p^−s

α

.

Ifϑ>0,thenTheorem1.3immediatelyimpliesthatan(Cϕα,ϑ)n^−ϑ(logn)^−2α1 forn≥ 2,seeCorollary8.1.Similarlytothecaseofaffinemapsdiscussedabove,Theorem1.3(a) does not provide the correct lower bound when ϑ = 0. In this case we shall instead proceed via the change of variable formula of Lemma6.2 and detailed analysis of the restrictedcountingfunction.

Theorem 1.6. For a positive integer c0 and a real number 0 < α < 1, let ϕα(s) = c0s+ Φα(p^−s).Thenϕα isin G_≥1 and

an(Cϕα)(logn)^α−12α forn≥2.

Inthe classical setting of H²(D), detailed studies of the approximation numbers of composition operators generated by symbols that map into an angle are carried out in [12] and [16]. Via the transference principle of [15, Sec. 9], these results also yield estimates for the approximation numbers of composition operators _Cψα:_H² → H² generatedbyanglemaps ψα(s)= 1/2+ Φα(p^−s).

Organization.InthepreliminarySection2wegivetheproofofTheorem1.1,anddiscuss thenotionofverticallimitfunctions.InSection3weanalyzetheorthogonaldecomposi- tionof CϕandproveTheorem1.2andTheorem1.3.InSection4weapplyTheorem1.3 toaffinesymbols,andinSection5tomembershipintheSchattenclasses.InSection6we introduceandstudyrestrictedcountingfunctionsandtheirassociatedStantonformulas.

InSection7weprovidetheproofofTheorem1.5.InSection8westudytheexampleof angle maps.

Notation. Wewill sometimesuse the notation f(x) g(x) to indicatethat there is a constant C such that f(x) ≤ Cg(x) for all relevant x. The notation indicates the reverse estimate,andf(x)g(x) meansthatf(x)g(x) andg(x)f(x).

Acknowledgments.Theauthorsthanktheanonymousrefereeforsuggestinganimprove- mentto Theorem1.3.

2. Preliminaries

We willhave usefortwo additional characterizationsoftheapproximation numbers of aboundedoperatorT onaHilbertspaceH,

an(T) = sup

E⊆H dim(E)=n

x∈Einf

x=1

T x, (2.1)

an(T) = inf

dim(E)=n−1E⊆H

sup

x∈E^⊥ x=1

T x. (2.2)

See forexample[9,Sec. II.7].Recall also thatapproximationnumberssatisfythe ideal property

a_n(S₁T S₂)≤ S₁a_n(T)S₂ (2.3) forbounded operatorsS1,T,andS2onaHilbertspaceH.

Thefollowing demonstration ofTheorem1.1,adaptedfrom[5], illustratestheuseof (2.1) and (2.3).Intheproof,we alsomakeuseofthefollowingresultfrom[10, p. 329].

Lemma 2.1.If ϕ∈G≥1,thenCϕ= 1.

Proof of Theorem1.1. Webeginwiththeupperboundin(1.2).Setψ(s)=s+ϑ.Note, bythedefinition(1.1) ofϑ,thatϕ−ϑisin_G≥1.Since_Cϕ−ϑCψ =_Cϕ,theidealproperty (2.3) withS₁=I,T =_Cϕ−ϑ, andS₂=_Cψ,thereforeyields

an(Cϕ)≤ Cϕ−ϑan(Cψ) =n^−ϑ,

wherethefinalequalityfollowsfromLemma2.1andthetrivialanalysisofCψpresented intheintroduction.

For the lower bound in (1.2), we choose E = span({e2,e3,. . . ,epn}) as the n- dimensional subspace of H² in (2.1). To estimate the infimum of CϕfH², for f(s)=n

j=1bjp^−s_j ofunitnorm, weconsidertheauxiliarysubspace F= span({e2^c0, e3^c0, . . . , e_p^c_n⁰})

anddeducefromthefundamentaltheoremofarithmetic,orthogonality,andtheCauchy–

Schwarzinequalitythat

CϕfH² ≥ sup

g∈F g_H2=1

Cϕf, gH²= _n

j=1

|bj|²p⁻_j^{2 Rec1 1}₂

.

Takingtheinfimumon theright-handside,over allf ∈E of unitnorm, weobtainthe statedlowerboundan(Cϕ)≥p⁻_n^Rec1.

Wewill now briefly recall afew factsabout vertical limit functionsand generalized boundaryvalues.LetT^∞denotethecountableinfiniteCartesianproductoftheunitcir- cleT inthecomplexplane,endowedwithitsHaarmeasureμ_∞.Viaprimefactorization, wemayviewanyχ∈T^∞ asacharacter,

χ(n) =χ^α₁¹χ^α₂²· · ·χ^α_d^d for n=

d

j=1

p^α_j^j.

ForaDirichletseriesf(s)=

n≥1b_nn^−sandacharacterχ∈T^∞,considerthevertical limitfunction

fχ(s) = ∞ n=1

bnχ(n)n^−s.

Iff convergesuniformly inCθ forsomeθ∈R,then {f_χ}χ∈T^∞ consistsprecisely ofthe functionswhichcanbeobtainedas uniformlimitsinCθ ofverticaltranslatesf(·+iτk), where (τk)k≥1 isasequence of real numbers.Despite thefactthata functionf ∈H² needonly converge inC_1/2, theDirichlet seriesfχ actuallyconverges inC0 foralmost everyχ∈T^∞ (seee.g. [11,Thm. 4.1]).Moreover,thegeneralizedboundaryvalue

f^∗(χ) = lim

σ→0⁺fχ(σ) existsforalmost everyχ∈T^∞,and

fH² =f^∗L²(T^∞). (2.4) Thefollowingresultcanbe extractedfrom [6,Sec. 2].

Lemma2.2. Supposethat ϕ:C0→C0 isaDirichletserieswhichconvergesuniformlyin Cε foreveryε>0.Then

(i) ϕ_χ(C₀)=ϕ(C₀)forevery χ∈T^∞,and (ii) ϕ^∗(χ)existsforalmosteveryχ∈T^∞.

Inparticular,wededucefromLemma2.2thatifϕ(s)=c0s+ϕ0(s) isin G,thenthe expression (1.1) forϑhasthereformulation

ϑ= ess inf

χ∈T^∞Reϕ^∗₀(χ). (2.5)

Following [10], we extend the notion of vertical limit functions to symbols ϕ ∈ G by defining

ϕ_χ(s) =c₀s+ (ϕ₀)_χ(s).

Theinteractionbetweenthecompositionoperator Cϕ andverticallimitsisgivenin[10, Prop. 4.3]:

(Cϕf)χ =Cϕχfχ^c0, (2.6) where f ∈ H², χ ∈ T^∞, and χ^c0(n) = χ(n)^c0 = χ(n^c0). Combining Lemma 2.2 (ii), (2.4),and (2.6) yieldsthefollowing result.

Lemma 2.3.If ϕ∈G and χ∈T^∞,then Cϕ andCϕχ areunitarilyequivalent.

3. Orthogonaldecompositionandapproximationnumbers

Wenow fixasubsetP ofthefullsetofprimenumbers.Foreachj∈M(P^⊥),welet H_j² denotethe subspaceof H² comprised ofDirichlet seriesof theform ejf, wheref is supportedonP.Since H_j²₁ ⊥H_j²₂ ifj1=j2,wehavetheorthogonaldecomposition

H²=

j∈M(P^⊥)

H_j². (3.1)

Thefollowing simpleobservationisthestartingpointofthepresentpaper.

Lemma3.1. Letϕ∈G_≥1 andsupposethat ϕ0issupportedonP.Forevery j∈M(P^⊥), let Cϕ,j denotetheoperatorobtainedby restricting Cϕ to H_j².Then

Cϕ=

j∈M(P^⊥)

Cϕ,j.

Proof. In view of (3.1), it is sufficient to prove thatCϕ maps H_j² to H_j^2c0, since the mapj→j^c0 isinjectiveon M(P^⊥).Considertheactionof Cϕ onen,wheren=jk for j∈M(P^⊥) andk∈M(P):

Cϕen(s) =j^−c0sk^−c0sn^−ϕ0(s).

Weseethat Cϕen ∈H_j^2c0, asaconsequenceof theassumptionthatϕ0 issupportedon P.

InviewofLemma3.1thereisforeveryn≥1 somem≥1 andj∈M(P^⊥) suchthat a_n(_Cϕ)=a_m(_Cϕ,j). Wefirstapply thisto obtainalower boundfor theapproximation numbersof _Cϕ whichwillimmediately implyourfirstmain result.

Lemma 3.2. Let ϕ ∈ G≥1 and suppose that ϕ0 is supported on a sparse set of prime numbersP.Thereisthenapositiveinteger m=m(P)such that

a_n(_Cϕ)≥ Cϕe_mnH².

Proof. Bydefinition,any fj ∈H_j² canbe writtenfj =ejf forafunctionf supported onP,andfjH² =fH².ByLemma2.2(ii)andthecompositionrule(2.6),wehave that

(Cϕfj)^∗(χ) =χ^c0(j)j^{−ϕ∗0(χ)}fχ^c0(ϕ^∗₀(χ))

foralmosteveryχ∈T^∞.Thisformulaisatfirstvalidforpolynomialsf,butbyadensity argumentitcontinuestoholdforgeneralf supportedonP,ifweinterpretfχ^c0(ϕ^∗₀(χ)) asageneralizedboundaryvaluewhenneeded.By(2.4) wethereforehavethat

Cϕfj²_H² =

T^∞

j^{−2 Reϕ∗0(χ)}|fχ^c0(ϕ^∗₀(χ))|² dμ_∞(χ).

In particular, j → Cϕ,j = a1(Cϕ,j) is decreasing for j ∈ M(P^⊥). Letting (jn)n≥1

denotetheincreasingsequenceofintegersin M(P^⊥),weconcludethat a_n(_Cϕ)≥a₁(_Cϕ,jn) =Cϕ,jn ≥ Cϕe_jnH²,

sinceejn∈H_j²_n andejnH² = 1.ThehypothesisthatP issparsemeansthat

N→∞lim 1 N card

j∈M(P^⊥) : j≤N

=

p∈P

1−1

p

=C(P)= 0,

andthusthatthereisapositiveintegermsuchthatjn≤mnforeveryn≥1.Therefore

Cϕejn²_H²≥

T^∞

(mn)^{−2 Reϕ∗0(χ)}dμ_∞(χ) =Cϕemn²_H².

Proof of Theorem1.2. Since ϕ₀ is supported on a sparse set of prime numbers, Lemma3.2yieldsthat

an(Cϕ)≥ CϕemnH²

for some positive integer m. Set Xε = {χ∈T^∞ : ϑ≤Reϕ^∗₀(χ)≤ϑ+ε}. Then μ_∞(Xε)>0,referringto(2.5),andaccordingly

Cϕemn²_H² =

T^∞

(mn)^{−2 Reϕ∗0(χ)}dμ_∞(χ)≥μ_∞(Xε)(mn)^−2(ϑ+ε).

This givesthestatedestimatewith C=

μ_∞(X_ε)m^−ϑ−ε. Wenow turntowardprovingTheorem 1.3(a).

Lemma 3.3. Suppose that ϕ∈G≥1 and let m be a positiveinteger. There isa constant C=C(ϕ0,m)>0suchthat

CϕenH²≤CCϕemnH²

forevery integer n≥1.

Proof. Asbefore,wecomputethenormsonT^∞,sothat Cϕe_n²_H²=

T^∞

n^{−2 Reϕ∗0(χ)}dμ_∞(χ).

Foranyε>0,considerthesetXε={χ∈T^∞ : ϑ≤Reϕ^∗₀(χ)≤ϑ+ε}.Asintheproof of Theorem1.2, weknowthatμ_∞(Xε)>0.Since x→n^−x isnon-increasing forx>0, itfollows, byinterpretingeachsideoftheinequalityas anaverage,that

Cϕe_n²_H² ≤ 1 μ_∞(Xε)

Xε

n^{−2 Reϕ∗0(χ)}dμ_∞(χ).

BythedefinitionofXε, wefindthat

Xε

n^{−2 Reϕ∗0(χ)}dμ_∞(χ)≤m^2(ϑ+ε)

Xε

(mn)^{−2 Reϕ∗0(χ)}dμ_∞(χ).

Extending thefinalintegralfromXεtoT^∞,weconcludethat

Cϕen²_H² ≤ m^2(ϑ+ε)

μ_∞(X_ε)Cϕemn²_H².

Proof of Theorem1.3(a). CombiningLemma3.2and Lemma3.3yieldsthat a_n(_Cϕ)≥ Cϕe_mnH²≥C⁻¹Cϕe_nH²,

wheremisas inLemma3.2andC isfrom Lemma3.3.

TheremainderofthissectionisdevotedtotheproofofTheorem1.3(b).Fornotational reasons,weintroducethepartial zetafunction

ζP(s) =

p∈P

1 1−p^−s.

ItisclearthatifP is ν-sparseforsome0< ν ≤1,thenζP(ν)<∞.

Lemma 3.4. Suppose that P is a set of ν-sparse prime numbers for some 0 < ν ≤ 1.

Then

k∈M(P) k≥K

k^−2σ ≤ζP(ν)K^ν−2σ

foreveryK∈M(P)andevery 2σ≥ν.

Proof. Weestimate

k∈M(P) k≥K

k^−2σ≤K^ν−2σ

k∈M(P) k≥K

k^−ν≤K^ν−2σ

k∈M(P)

k^−ν =K^ν−2σζP(ν).

Lemma 3.5. Fix ϕ ∈G≥1 and suppose that ϕ₀ issupported on a ν-sparse set of prime numbersP forsome0< ν≤1.If2ϑ≥ν,then

am(Cϕ,j)≤

ζP(ν)k_m^ν/2CϕejkmH²,

where(k_m)_m≥1 are theintegersof _M(P)inincreasing orderand j∈M(P^⊥).

Proof. Weapplythemin-maxprinciple(2.2),choosingE⊆H_j² as E= span

ejk1, ejk2, . . . , ejk_m−1

.

Thisgivesusthat

am(Cϕ,j)≤ sup

f∈E^⊥ f_H²=1

CϕfH².

Accordingly, suppose thatf ∈E^⊥ with fH² = 1. IfRes ≥ϑ, theCauchy–Schwarz inequalityandLemma3.4implythatf(s) convergesabsolutely, andthat

|f(s)|²≤

k∈M(P) k≥km

(jk)^{−2 Res}=j^{−2 Res}

k∈M(P) k≥km

k^{−2 Res}≤ζP(ν)k^ν_m(jkm)^{−2 Res}.

Of course thesame estimatealso holds iff isreplaced by f_χ^c0 for any χ∈T^∞. Since s= Reϕ^∗₀(χ)≥ϑforalmosteveryχ,wemaythereforeapplythisestimateinconjunction with Lemma2.2(ii), (2.4) and(2.6) toseethat

Cϕf²_H²=

T^∞

|f_χ^c0(ϕ^∗₀(χ))|²dμ_∞(χ)

≤

T^∞

ζ_P(ν)k^ν_m(jk_m)^{−2 Reϕ∗0(χ)}dμ_∞(χ) =ζ_P(ν)k_m^νCϕe_jkm²_H².

Togetherwith themin-maxprinciple,this givestheclaimedestimate.

Proof of Theorem1.3 (b). ThefunctionΦ : [1,∞)→(0,1] definedby

Φ(x) =

⎛

⎝

T^∞

x^{−2 Reϕ∗0(χ)}dμ_∞(χ)

⎞

⎠

1 2

isstrictlydecreasing,onto(bytheassumptionϑ>0),continuousandenjoystheestimate Φ(xy) ≤ y^−ϑΦ(x) for every x,y ≥ 1. Hence Φ has an inverse function Φ⁻¹: (0,1] → [1,∞) satisfyingthesamepropertiesandenjoyingtheestimate

Φ⁻¹(xy)≤y^−1/ϑΦ⁻¹(x) (3.2) forevery0< x,y≤1.Fixsome0< x≤1.TheorthogonaldecompositionofLemma3.1 allowsusto rewrite

n∈N : a_n(_Cϕ)≥

ζ_P(ν)x=(j, m)∈M(P^⊥)×N : a_m(_Cϕj)≥

ζ_P(ν)x. We now apply Lemma 3.5 to bound the right-hand side from above. Note that the hypotheses of Lemma3.5 certainly hold, since we are working under the stronger as- sumptionsthat0< ν <1 and2ϑ≥ν/(1−ν).Weobtainthat

n∈N : a_n(_Cϕ)≥

ζ_P(ν)x

≤(j, m)∈M(P^⊥)×N : k^ν/2_m Φ(jk_m)≥x

=(j, m)∈M(P^⊥)×N : j≤Φ⁻¹(xk_m^−ν/2)/k_m.

Countingforeachmthenumberofpositiveintegersj(notonlythoseinM(P^⊥))which satisfytheinequalityj≤Φ⁻¹(xkm^−ν/2)/km,wetherefore havetheupperbound

n∈N : a_n(_Cϕ)≥

ζP(ν)x≤ ^∞

m=1

Φ⁻¹ xkm^−ν/2

k_m ≤Φ⁻¹(x)ζP(1−ν/(2ϑ)), where thesecond inequalitycomes from (3.2) applied with y =km^−ν/2 ≤1.Since 2ϑ≥ ν/(1−ν),weconcludethattheestimate

n∈N : a_n(_Cϕ)≥

ζ_P(ν)x≤ζ_P(ν) Φ⁻¹(x) (3.3) holdsforevery0< x≤1.

Since ϑ >0, there is a smallest positive integer N suchthat N^2ϑ ≥ ζP(ν). By the upperboundinTheorem1.1itfollowsthata_n(_Cϕ)≤

ζP(ν) foreveryn≥N.Applying (3.3) withx=a_n(_Cϕ)/

ζ_P(ν)≤1 immediatelygivesusthat an(Cϕ)≤

ζP(ν)Φ n

ζP(ν)

(3.4) foreveryn≥N.Following theproof ofLemma3.3verbatim withε=ϑyieldsthat

Φ n

ζ_P(ν)

≤

ζP(ν)2ϑ

μ_∞(X)Φ(n), (3.5)

where the set X = {χ∈T^∞ : ϑ≤Reϕ^∗₀(χ)≤2ϑ} satisfies μ_∞(X) > 0. Combining (3.4) and(3.5),weconcludethat

an(Cϕ)≤

ζP(ν)1/2+2ϑ

μ_∞(X) CϕenH²

foreveryn≥N,whichcompletestheproof.

4. Compositionoperatorsgeneratedbyaffinesymbols

Toexemplify Theorem 1.3we consider affine symbols, whichwe recallfrom (1.3) to havetheform

ϕ(s) =c0s+c1+

p∈P

cpp^−s.

At first, we assume that ϕ is supported by a set of |P| = d < ∞ prime numbers. In particular,cp= 0 foreveryp∈P.Notefrom (2.5) that

ϑ= Rec1−

p∈P

|cp|.

Beforeproving Corollary1.4,letusquicklyrecalltheknownresultsabouta_n(_Cϕ) in this setting.Webegin withthecasec₀= 0,inwhichcasewemustrequirethatϑ≥1/2 inorder forCϕ tobe bounded. QueffélecandSeip[15,Thm. 1.3] have establishedthat ifϑ= 1/2,then

1 n

(d−1)/2

a_n(_Cϕ) logn

n

(d−1)/2

.

Ifϑ>1/2,thenby[13,Thm. 4.1] we havethat an(Cϕ)

Rec1−ϑ Rec1−1/2

n

,

where the impliedconstantdepends onRec1 andϑ>1/2,butnot ond. Actually, the estimate is stated and proved only for d = 1 in [13]. However, it can be extended to general d≥1 byapplying themax-minprinciple (2.1) andthe subordinationprinciple foraffinesymbols from[6,Thm. 5].

Supposeinsteadthatc₀≥1.Ifϑ>0,thenthebestpreviouslyknownestimateswere from Theorem 1.1. As mentioned inthe introduction, ifϑ = 0 for anaffine symbol ϕ, then an(Cϕ)1 forn≥1,andso thiscaseisnotof interest.

To proveCorollary1.4, werequirethefollowing versionof Hankel’sasymptoticesti- mate for the modified Bessel function of the second kind with parameter 0. It will be convenientforustohaveexplicitconstants;wehavemadenoattempttooptimizethese.

Lemma 4.1.If x≥ ¹₈,then 1 π√

2e

√1 x ≤

π

−π

e^{−4xsin2(θ/2)}dθ 2π ≤

√π 4

√1 x.

Proof. Fortheupperbound,weusethat|sin(θ/2)|≥ |θ/π|for−π≤θ≤πtoconclude that

π

−π

e^{−4xsin2(θ/2)}dθ 2π ≤

∞

−∞

e^−4xπ2θ2 dθ 2π =

√π 4

√1 x. Forthelower bound,wesupposethat0< ε≤2√

x.Then π

−π

e^{−4xsin2(θ/2)} dθ

2π ≥e^−ε2 2π

−π≤θ≤π : |sin(θ/2)| ≤ ε 2√ x

≥εe^−ε2 π

√1 x,

whereweusedthat|sin(θ/2)|≤ |θ/2|forthefinalinequality.Thestatedlowerboundis obtainedbychoosingε= 1/√

2,whichispermissiblebytheassumptionthatx≥1/8.

Proof of Corollary1.4. InviewofLemma2.3wemaywithoutlossofgeneralityreplace ϕbyϕ_χ foranyχ∈T^∞This allowsus toassumethatϕ₀ isoftheform

ϕ₀(s) =ϑ+iτ+

p∈P

γ_p(1−p^−s),

whereτ ∈Randγp>0 for p∈P.ByTheorem 1.3(a)and(b),weneed toestimate

Cϕen²_H² =

T^∞

n^{−2 Reϕ∗0(χ)}dμ_∞(χ) =n^−2ϑ

p∈P

π

−π

n^{−2γp(1−cosθp)}dθp

2π

as n→ ∞.Supposethatn islargeenoughthatγplogn≥ ¹₈ foreveryp∈P.Then, by applyingLemma4.1withx=γplogn,

p∈P

π

−π

n^{−2γp(1−cosθp)}dθp

2π =

p∈P

π

−π

n^{−4γpsin2(θp/2)}dθp

2π (logn)^−d2. Wefinish this sectionby discussing aclassof affine symbolswith |P| =∞. Forany affine symbol with absolutely convergentcoefficients, the image ϕ^∗₀(T^∞) is anannulus (seee.g. [19,Sec. XI.5]).Henceϕ^∗₀(T^∞) touchesthelineRew=ϑtangentially.However, theexamplesof thissectionshowthattheinteraction betweendifferentprimenumbers is essential in determining the behavior of the approximation numbersa_n(Cϕ). When c₀= 0,symbolswith|P|=∞havepreviouslybeenconsideredin[15,Thm. 1.3].

Theorem 4.2. Let P = (pj)_j≥1 be a set of prime numbers which is ν-sparse for every ν >0.Forfixedc0≥1,ϑ>0,andβ >1,define

ϕ(s) =c0s+ϑ+ ∞ j=1

1−p⁻_j^s j^β .

Thenthereare positiveconstantsC1=C1(β)andC2=C2(β)suchthat n^−ϑe^{−C1(logn)1/β} an(Cϕ)n^−ϑe^{−C2(logn)1/β} forn≥2.

Proof. Since P is ν-sparse for everyν > 0 andsince ϑ >0, we canuse both parts of Theorem1.3to concludethat

(an(Cϕ))² Cϕen²_H² =n^−2ϑ

∞ j=1

π

−π

n⁻

2

jβ(1−cosθj)dθj

2π. Weneedto estimatetheintegrals

I_j,β(n) = π

−π

n^{−jβ2(1−cosθj)}dθj

2π

for n ≥ 2. Let J = (logn)^1/β. When j > J we estimate roughly to obtain that n^−4/jβ ≤Ij,β(n)≤1.Hence

exp

− 4

β−1(logn)^1/β

≤

j>J

I_j,β(n)≤1. (4.1)

Nextweturn to1≤j ≤J,applying Lemma4.1with x= (logn)/j^β to seethat 1

π√ 2e

J J j=1

j^β logn ≤

J

j=1

I_j,β(n)≤ √

π 4

J J j=1

j^β

logn. (4.2) From Stirling’sformulawe findthat

J

j=1

j^β

logn exp β

2

J+1 2

log(J)−J

−J

2log logn

.

Thatis,sinceJ =(logn)^1/β,

J

j=1

j^β

logn exp

−β

2(logn)^1/β+log logn 4

. (4.3)

Combining (4.1), (4.2), and (4.3), notingthat ¹

π√

2e < ^√₄^π <1, yields thedesired esti- mates.

5. Schattenclasses

For 1 ≤q < ∞, alinear operator T on a Hilbert space H belongs to the Schatten class S_qif(a_n(T))_n≥1∈^q,inwhichcaseitsSchatten normisgivenby

T^qSq = ∞ n=1

|an(T)|^q.

Let(xn)n≥1be anorthonormalbasisofH.IfT ∈Sq,then