Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle

We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardy space $\mathcal{H}^2$ of Dirichlet series. For $p\geq 2$, these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for $\mathcal{H}^2$ and we give examples of bounded composition operators with Dirichlet series symbols on $\mathcal{H}^p,\,p>0$.


Introduction
The Hardy space H 2 of Dirichlet series, which was first systematically studied by H. Hedenmalm, P. Lindqvist, and K. Seip [13], is defined as Gordon and Hedenmalm [12] determined the class G of symbols which generate bounded composition operators on the Hardy space H 2 .The Gordon-Hedenmalm class G consists of all functions ψ(s) = c 0 s + ϕ(s), where c 0 is a non-negative integer, called the characteristic of ψ, and ϕ is a Dirichlet series such that: (i) If c 0 = 0, then ϕ(C 0 ) ⊂ C 1  2 .(ii) If c 0 ≥ 1, then ϕ(C 0 ) ⊂ C 0 or ϕ ≡ iτ for some τ ∈ R.
We denote by C θ , θ ∈ R the half-plane {s : Re s > θ}.We will also use the notation G 0 and G ≥1 for the subclasses of symbols that satisfy (i) and (ii), respectively.
In this paper, we are mostly interested in the case ψ = ϕ ∈ G 0 .In that context, the compact operators C ϕ : H 2 → H 2 were characterized only very recently in [9], in terms of the behavior of the mean counting function Re s, w ∈ C 1 2 \{ϕ(+∞)}.

It turns out that C ϕ is compact if and only if (1) lim
Re w→ The next step would be to characterize symbols ϕ ∈ G 0 such that C ϕ belongs to the Schatten class S p , p > 0. In the disk setting D. H. Luecking and K. Zhu [17] proved that a composition 1 operator C φ on the Hardy space H 2 (D) belongs to the Schatten class S p , p > 0 if and only if (2) where φ is a holomorphic self-map of the unit disk and N φ is the associated Nevanlinna counting function [24].
Our first main result is that the analogue characterization holds in the Dirichlet series setting provided the symbol has bounded imaginary part.
Theorem 1.1.Suppose that the symbol ϕ ∈ G 0 has bounded imaginary part and that p ≥ 1.Then, the composition operator C ϕ belongs to the class S 2p if and only if ϕ satisfies the condition For p > 0 the above condition remains necessary and if p ≥ 2 it is necessary for all symbols in G 0 .
When p = 1, namely if we want to know if C ϕ is Hilbert-Schmidt, things are easier and Hilbert-Schmidt composition operators with symbols ϕ in G 0 have already been characterized in [9].This is equivalent to saying that Our next result is a comparison-type principle.Using the Lindelöf principle for Green's functions, we will be able to establish geometric conditions on the symbols that imply that the associated composition operator is compact or belongs to S p .To our knowledge this is the first example of a technique that gives geometric conditions that apply to all symbols ϕ ∈ G 0 .To exemplify this, we focus on symbols whose range is contained in angular sectors.
We can strengthen the previous result proving that if the range of the symbol meets the boundary inside a finite union of angular sectors, then the induced composition operator is compact.
This geometric method also applies to continuity and compactness of composition operators acting on the other Hardy space of Dirichlet series H p , p = 2. Recall that for 0 < p < ∞, the Hardy space H p of Dirichlet series is defined as the completion of Dirichlet polynomials under the Besicovitch norm (or quasi-norm if 0 < p < 1) The characterization of bounded composition operators with Dirichlet series symbols on H p , p / ∈ 2N is an open and challenging question.The condition ϕ ∈ G 0 is necessary but not sufficient, [21] and there is no known sufficient conditions which may be applied to a large class of symbols whose range touches the boundary of C 0 .We provide such a sufficient condition under the assumption that the range of the symbol is contained in an angular sector.
then the composition operator is compact on H q .
In the last section we briefly discuss the case of Bergman spaces of Dirichlet series as well as some results on Carleson measures.
Notation.Throughout the article, we will be using the convention that C denotes a positive constant which may vary from line to line.We will write that C = C(x) to indicate that the constant depends on a parameter x.If f, g are two real functions defined on the same set Ω, we will write f ≪ g if there exists C > 0 such that for all x ∈ Ω, f (x) ≤ Cg(x) and f ∼ g if f ≪ g and g ≪ f .
Acknowledgments.We thank Ole Fredrik Brevig and Karl-Mikael Perfekt for providing helpful comments.
Part of the work has been conducted during research visits of the second author at Université Clermont Auvergne and Lund University (supported from the project Pure Mathematics in Norway -Ren matematikk i Norge funded by the Trond Mohn Foundation).He wants to express his gratitude toward Alexandru Aleman and the previously mentioned departments for their hospitality.

Schatten classes.
A compact operator T acting on a separable Hilbert space H can be written as ( 5) where {s n } n≥1 is the sequence of singular values and {e n } n≥1 and {h n } n≥1 are orthonormal sequences.In case T is self-adjoint, then e n = h n for all n ≥ 1.For p > 0 the S p Schatten class of compact operators T on H is defined as Equivalently (see [14]), for p ≥ For a positive operator T on H we define the power T p , p > 0, as When p = n ∈ N, the operator T n is the n-th iteration of T .We observe that T ∈ S p if and only if T p ∈ S 1 .It T is not assumed to be positive, we can still use that For a unit vector x ∈ H and a positive operator T , applying Hölder's inequality in (5) we obtain the following inequality (6) For 0 < p ≤ 1 the inequality is reversed.
2.2.The infinite polytorus and vertical limits.The infinite polytorus T ∞ is defined as the (countable) infinite Cartesian product of copies of the unit circle T, It is a compact abelian group with respect to coordinatewise multiplication.We can identify the Haar measure m ∞ of the infinite polytorus with the countable infinite product measure m × m × • • • , where m is the normalized Lebesgue measure of the unit circle.
T ∞ is isomorphic to the group of characters of (Q + , •).Given a point χ = (χ 1 , χ 2 , . . . ) ∈ T ∞ , the coresponding character χ : Q + → T is the completely multiplicative function on N such that χ(p j ) = χ j , where {p j } j≥1 is the increasing sequence of primes, extended to an n s is a Dirichlet series and χ(n) is a character.The vertical limit function f χ is defined as By Kronecker's theorem [6], for any ǫ > 0, there exists a sequence of real numbers {t j } j≥1 such that f (s + it j ) → f χ (s) uniformly on C σu(f )+ǫ , where σ u (f ) denotes the abscissa of uniform convergence of f.If f ∈ H 2 , then for almost every character χ ∈ T ∞ the vertical limit function f χ converges in the right half-plane and has boundary values f χ (it) = lim σ→0 + f χ (σ + it) for almost every t ∈ R, [2].For ψ(s) = c 0 s + ϕ(s) ∈ G, we set The symbol ψ has boundary values ψ χ (it) = lim σ→0 + ψ χ (σ + it) for almost every χ ∈ T ∞ and for almost every t ∈ R.
where ϕ ∈ G 0 and f ∈ H 2 .By f (+∞) we denote the first coefficient a 1 of the Dirichlet series f (s) = n≥1 an n s .We apply the polarization identity in (8) yielding to We will make use of two properties of the counting function M ϕ (w) proved in [9], the submean value property and a Littlewood type inequality.Those respectively are (10) M for every disk D(w, r) ⊂ C 1 2 that does not contain ϕ(+∞), and In Subsection 4 we will prove a weaker version of the Littlewood inequality (11) but sufficient for our purpose.The standard technique to prove such inequalities goes through regularity results for conformal maps [9,15].We shall use the following consequence of (11) (see [9,Lemma 2.3]): for σ ∞ > Re(ϕ(+∞)), there exists C > 0 such that, for all w ∈ C σ∞ ,

Carleson measures.
Let H be a Hilbert space of holomorphic functions on a domain Ω.
A Borel measure µ in Ω is called a Carleson measure for H if there exists a constant C > 0 such that, for all f ∈ H, H .We will denote by C(µ, H) or simply by C(µ) the infimum of such constants.For instance, Carleson measures on the Hardy space H 2 (C 1  2 ), that consist of holomorphic function are characterized as follows.
2 ) if and only if there exists a constant D > 0 such that for every square Q with one side I on the line Moreover, there exist two absolute constants a, b > 0 such that, for all Borel measures µ on C 1 2 , denoting by D(µ) the infimum of the constants D verifying (13), then aD(µ) ≤ C(µ) ≤ bD(µ).

2.5.
Weighted Hilbert spaces of Dirichlet series.Our main strategy (inspired by [17]) to obtain the membership of C ϕ to S 2p is to derive it from the membership to S p of an associated Toeplitz operator defined on another space of Dirichlet series.We now introduce this class of spaces.For a ≤ 1 we define the weighted Hilbert space D a of Dirichlet series as The reproducing kernel k w,−a , a ≥ 0 of (D −a ) 0 (space mod constants) at a point w ∈ C 1 2 is given by ( 14) where The local embedding theorem, [13], states that there exists an absolute constant C > 0 such that for every f ∈ H 2 (15) 1 A direct application of ( 15) is that for every f ∈ (D −a ) 0 , a > 0, we have the following embedding In particular, if A is a subset of C 1 2 with bounded imaginary part, then The differentiation operator D(f ) = f ′ is an isometry between H 2 0 and (D −2 ) 0 .By (9) the composition operator C ϕ belongs to S 2p (H 2 ), p > 0 if and only if b) Assume that there exists ρ : ϕ(C 0 ) → (0, +∞) such that ρdA is a Carleson measure for (D −2 ) 0 and that (18) ϕ(C0) Proof.We start by proving a).Let I µ be the inclusion operator from (D −2 ) 0 into L 2 (C 1 2 , µ) which is bounded since µ is Carleson.Moreover, assuming C ϕ ∈ S 2p or, equivalently, T ϕ ∈ S p , we get by the ideal property of Schatten classes that by ( 6), where K w,−2 is the normalized reproducing kernel of (D −2 ) 0 at w. Observe that the exchange of integral and sum is justified by Tonelli's theorem.Fix σ ∞ > Re ϕ(+∞).By (17), By (14) one can estimate the behaviour of K w,−2 (z) in the disc D w, where the last inequality follows from the submean value property of the mean counting function (10).Taking into account the value of k w,−2 we get Finally, (12) yields Conversely, assume that (18) holds and let q be the conjugate exponent of p.Let (f n ) be any orthonormal basis of (D −2 ) 0 .Then Since ρdA is a Carleson measure and since n |f n (w)| 2 = k w,−2 (w) for any orthonormal basis of (D −2 ) 0 , we get Hence, T ϕ belongs to S p .
In view of the above theorem, the ideal case would be to choose a function ρ : ϕ(C 0 ) → (0, +∞) such that ρdA is a Carleson measure for (D −2 ) 0 and This yields to ρ(w) = Re(w) − 1 2 .Now if ϕ has bounded imaginary part, then the embedding inequality implies that 1 ϕ(C0) Re(w) − 1 2 dA is a Carleson measure for (D −2 ) 0 .This gives the way to the case p > 1 of Theorem 1.1.
Corollary 3.2.Let p > 1 and ϕ ∈ G 0 with bounded imaginary part.Then C ϕ belongs to S 2p if and only if Proof.Our discussion actually shows that, under the assumptions of the corollary, C ϕ ∈ S 2p if and only if We may conclude if we are able to prove that for any ϕ ∈ G 0 , Both of these properties follow from (12).
When ϕ does not have bounded imaginary part, there are still interesting Carleson measures for (D −2 ) 0 , for instance Re(w) − 1 2 /(1+| Im(w)|) a dA for any a > 1.This yields to the following result.
Proof.This follows from Theorem 3.1 with ρ(w) = Re(w) − We now prove that (3) remains necessary for p ≥ 2 without any assumption on ϕ.Proof.For the positive operator T ϕ belonging to S p , denoting by (f n ) an orthonormal basis of eigenvectors of T ϕ , The quantity under the integral sign is nonnegative since An application of Tonelli's theorem yields We now use ( 6) We conclude as above using the submean value property of the counting function (10) to deduce that (3) holds true.
We end up the proof of Theorem 1.1 by considering the case p ∈ (0, 1).Proof.We still denote by (f n ) an orthonormal basis of eigenvectors of T ϕ .We now write Now by ( 16) the measures dA are finite measures on C 1 2 with uniformly bounded mass.It follows from Hölder inequality that 3.2.The case of even integers.We now prove the 2m-Schatten class characterization (4).
Proof of Theorem 1.2.We first prove that (4) implies that C ϕ is compact.If this was not the case, then we could find δ > 0 and a sequence (w(k)) ⊂ C 1 2 with real part going to 1/2 such that for every ε ∈ (0, 1) the rectangles are pairwise disjoint and for all k ≥ 1, We recall that ζ ′′ (s) has a pole of order 3 at s = 1, thus we can choose ε > 0 close to zero such that , for every w = (w 1 , . . ., w m ) ∈ A k .Using the mean-value property of the counting function as well as the estimate above, we obtain that Since the sets A k are pairwise disjoint, this would contradict (4).
Hence, for both implications of Theorem 1.2, we may assume that C ϕ hence T ϕ is compact.Let us consider the canonical decomposition of T ϕ , T ϕ (f ) = n≥1 s n f, f n f n .We know that We observe that n≥1 Arguing as in the proof of Theorem 3.4, we may use Tonelli's theorem to get .
Proof.The norm of the image of a function f ∈ H 2 under C ϕ can be written as where µ ϕ if the push-forward measure of m ∞ by ϕ(χ), see [8].Since C ϕ is compact and the reproducing kernel we can argue like in the proof of [19,Theorem 3] to deduce that where Q is a (Carleson) square in C 1 2 with one side I on the vertical line Re s = 1 2 .This means that µ ϕ is a vanishing Carleson measure for H 2 (C 1  2 ) and this implies that µ ϕ {Re s= 1 2 } is absolutely continuous with respect to the Lebesgue measure of R. Following a standard argument, see for example [11, Chapter 3], we will prove that µ ϕ {Re s= 1 2 } is equal to 0. By the Lebesgue-Radon-Nikodym theorem there exists a positive function f ∈ L 1 (R) such that 2 } is of full measure if and only if f ≡ 0. Let us assume that there exists ε > 0 such that f −1 ((ε, ∞)) > 0. Let F ⊂ f −1 ((ε, +∞)) with positive and finite measure and let δ > 0 be such that for every Carleson square in C 1 2 with length |I| ≤ δ.We can cover F by a sequence of intervals We are now ready to give an analogue of Theorem 1.2 involving the symbol directly.
Theorem 3.7.Suppose that the symbol ϕ ∈ G 0 induces a compact operator and let m ∈ N.
Then, C ϕ belongs to S 2m if and only if (20) Proof.Let T = C * ϕ • C ϕ and let us consider its canonical decomposition We know that C ϕ ∈ S 2m ⇐⇒ T m ∈ S 1 and that As in the proof of Theorem 1.2 the quantity inside the integral is nonnegative which allows us to use Tonelli's theorem.Hence By induction one finally obtains dm ∞ (χ k ).

A comparison-type principle
4.1.Lindelöf principle and Littlewood inequality.In this section we will use Lindelöf principle for Green's function to give a simple proof of a non-contractive Littlewood inequality (11).Similar techniques have been used in the disk setting, [5].We recall (see for instance [22]) that a Green's function for a domain Ω ⊂ C is a function g Ω : Ω × Ω → (−∞, +∞] such that, for all w ∈ Ω, g(•, w) is harmonic in Ω\{w}, g Ω (z, w) → 0 n.e as z → ∂Ω and g Ω (•, w) + log | • −w| is harmonic in a neighbourhood of w.If a domain admits a Green's function then it is necessarily unique.For instance, the Green's function on the disk g D : D × D → (0, +∞] has the form By conformal invariance we can easily define Green's function on every simply connected subdomain of the complex plane, for example The class of domains D possessing a Green's function g D is much larger than the simply connected domains, see [22,Chapter 4].Lindelöf principle for Green's function (see for instance [4]) states that if f is a holomorphic function mapping D 1 to D 2 , where both of those domains possess Green's function, then for z 0 ∈ D 1 and w ∈ D 2 \ {f (z 0 )} ( 21) Let us first show how to deduce, up to a multiplicative constant, Littlewood inequality (11) and also a corresponding inequality for a symbol in G ≥1 (such an inequality was used in [3] to obtain a sufficient condition for composition operators with symbols in G ≥1 to be compact on H 2 ).Recall that for ψ ∈ G ≥1 , its restricted Nevanlinna counting function is defined by Proof.Let ϕ ∈ G 0 and w ∈ C 1 2 \{ϕ(+∞)}.For T > 0 sufficiently large, w = ϕ(T ) so that Lindelöf principle implies On the other hand, using the elementary inequality log(x) ≥ 1 2 (1 − x −2 ) valid for x > 0, Observe that {Re(s) : s ∈ ϕ −1 ({w})} is bounded.Therefore, for all ε ∈ (0, 1), we can choose T large enough so that We can conclude by letting T to +∞ and ε to 0. Regarding b), let χ ∈ T ∞ and w ∈ C 0 with Re w ≤ c 0 .Lindelöf principle says that Finally it was shown in [3] that where C does not depend neither on χ ∈ T ∞ nor on w with Re w ≤ c 0 .

4.2.
A comparison-type principle and a polygonal compactness theorem.We shall now apply the idea of the previous subsection when ϕ ∈ G 0 maps C 0 into a subdomain D of C 1 2 .Lindelöf principle helps us to find better estimates on M ϕ .Indeed, provided D admits a Green's function, the proof of Theorem 4.1 shows that (22) M ϕ (w) ≪ g D (w, ϕ(+∞)), w ∈ C 1 2 \{ϕ(+∞)}.We deduce the following comparison principle.Under similar conditions a norm-comparison principle appeared in [8].
. By T -periodicity of ϕ D , we know that by equality in Lindelöf principle for a Riemann map.Hence our assumption on C ϕD gives an estimate on M ϕD which transfers to M ϕ thanks to (22) which itself gives the corresponding result on C ϕ .Observe that in both cases, we use the characterization of compactness or membership to S 2p .
Remark.In Theorem 4.2, we can only assume that D admits a Green's function and use for R D a universal covering map of D.
The most interesting case occurs when ϕ(C 0 ) is mapped into an angular sector contained in C 1  2 .This leads to Theorem 1.3 that we now prove.
Using properties of conformal maps, we can extend the compactness part of Theorem 1.3 to slightly more general domains.This is the analogue of the polygonal compactness theorem, [25], in our setting.
Remark.Our techniques apply also for symbols ψ = c 0 s + ϕ ∈ G ≥1 .Although, the range of such a symbol cannot meet the imaginary axis in an angular sector or more generally inside a domain D where Im w is bounded for w ∈ D (C 0 \ C ε ), ε > 0. If that was the case then we would be able to find a point s(ε) such that Re ϕ (s(ε)) ≤ ε/4.The Dirichlet series ϕ converges uniformly in C Re s(ε)/2 , [7].By almost periodicity we can find an increasing unbounded sequence of positive numbers {T n } n≥1 such that Re ϕ(s(ε) We observe that |Im ψ(s(ε) + iT n )| → +∞, this contradicts our assumption.

4.3.
On the boundedness on H p .We conclude this section by the proof of Theorem 1.4.We will use Hilbertian methods to prove that our assumption implies that C ϕ is bounded as an operator from H to H 2 , where H is a Hilbert space of Dirichlet series containing H p .To do this, we need another class of Bergman spaces of Dirichlet series, the spaces A a , α ≥ 1.They are defined as Proof of Theorem 1.4.Let us set α = 2k/p.Let ε > 0, C > 0 be such that Re w ∈ (0, ε) implies Let us write f k = j≥1 a j j −s .By the Cauchy-Schwarz inequality, for all w ∈ C 1 2 +ε , By the local embedding theorem (23), the boundedness of pointwise evaluation at ϕ(+∞) and the continuity of C ϕ on H 2 , applied to 2 −s , we get Now, the inclusion operator i : H p/k → A α is contractive, [16].Therefore Let us turn to compactness.Let {f n } n≥1 be a sequence of H q converging weakly to 0. We set g n = f k n and observe that (g n ) converges pointwise to 0 on C 1 2 and that the Dirichlet coefficients g n (j) converge to 0 for each j ≥ 1.
We work as above but we now set α = 2k/q and consider δ ∈ (0, ε).Then The first term goes to zero as n tends to +∞ and the second term is as small as we want for every n if we adjust δ small enough.Therefore it remains to show that, for a fixed δ > 0, the last terms tends to 0 as n tends to +∞.Now, for all n ≥ 1 and all w ∈ C 1 2 +δ , Since (g n ) is bounded in A α , for any η > 0, there exists n 0 ∈ N such that, |g ′ n (w)| ≤ η|2 −w |.We now argue as above to conclude that (C ϕ (f n )) tends to 0 in H q .
Remark.We choose to work with symbols with range into angular sectors for the sake of simplicity.It will be interesting to know if our techniques can be applied to give other examples of geometric conditions related to the behavior of composition operators on Hardy spaces of Dirichlet series.If we further assume that ϕ has bounded imaginary part, then C ϕ belongs to the class S p , p ≥ 2 if and only if ϕ satisfies (24), and for p > 0 the condition remains necessary.

Further discussion
To prove Theorem 5.1 one can argue in a similar manner with the Hardy space H 2 , using the analogue key ingredients, those are: The change of variables formula [15, Theorem 1.2], the Littlewood-type inequality [15,Proposition 5.4], the weak submean value property [15,Theorem 4.11] and the behavior of reproducing kernels (14).5.2.Carleson measures.E. Saksman and J-F.Olsen [19] proved that if µ is a Carleson measure for H 2 , then it is a Carleson measure for H 2 (C 1 2 ).The converse is also true with the extra assumption that µ has compact support.
A direct consequence of the local embedding theorem is that a sufficient condition for a measure µ in 1 2 < Re s < σ ∞ to be Carleson for H 2 is {C(µ n , H 2 (C 1 2 ))} n∈Z ∈ ℓ 1 , where µ n is the restriction of µ on the half-strip {s ∈ C 1 2 : n ≤ Im s < n + 1}.Indeed, An example of such a measure is the restriction of Mϕ(w) Re w− 1 2 dA(w) to 1 2 < Re s < 1+Re ϕ(+∞)

2
. The above condition is not necessary, as we will exemplify now.
We consider the sequence {s n } n≥1 , where As we will prove in a moment the measure dµ(w) = Let Q n , n ≥ 1 be the square with center at the point s n and one side I n on the line {Re s = 1 2 }.Then, for some a ∈ (0, 1).We follow an argument of [23,Section 4], see also [1].It is sufficient to prove that the matrix A = defines a bounded operator on ℓ 2 .We will prove that for every j ∈ N

Theorem 1 . 4 .
Let k ∈ N and p ∈ (0, 2k].If the symbol ϕ ∈ G 0 maps the right half-plane into an angular sector of the form

2. 3 .
Composition operators on H 2 .O. F. Brevig and K-M.Perfekt [9, Theorem 1.3.]proved the following analogue of Stanton's formula for the Hardy spaces of Dirichlet series:

3 . 1 .Theorem 3 . 1 . 1 2( 2 p
Composition operators belonging to Schatten classes 3.Schatten class and Carleson measures.We shall divide the proof of Theorem 1.1 into several parts.We first handle the case p > 1 in a more general context by giving a necessary and a sufficient condition for C ϕ to belong to S 2p .Both conditions involve M ϕ and Carleson measures.At this stage, we do not assume anything on the imaginary part of ϕ.Let p > 1 and ϕ ∈ G 0 .a) Assume that C ϕ ∈ S 2p and let µ be a Carleson measure for (D −2 ) 0 .Then C M ϕ (w)) p ζ ′′ (2 Re(w)) Re(w) − 1 dµ(w) < +∞.