Trace ideal criteria for embeddings and composition operators on model spaces
aa r X i v : . [ m a t h . F A ] J u l TRACE IDEAL CRITERIA FOR EMBEDDINGS ANDCOMPOSITION OPERATORS ON MODEL SPACES
ALEXANDRU ALEMAN, YURII LYUBARSKII, EUGENIA MALINNIKOVA,KARL-MIKAEL PERFEKT
Abstract.
Let K ϑ be a model space generated by an inner function ϑ . We study the Schatten class membership of embeddings I : K ϑ ֒ → L ( µ ), µ a positive measure, and of composition operators C ϕ : K ϑ → H ( D ) with a holomorphic function ϕ : D → D . In the case of one-component inner functions ϑ we show that the problem can be reducedto the study of natural extensions of I and C ϕ to the Hardy-Smirnovspace E ( D ) in some domain D ⊃ D . In particular, we obtain a charac-terization of Schatten membership of C ϕ in terms of Nevanlinna count-ing function. By example this characterization does not hold true forgeneral ϑ . Introduction
Let D = { z : | z | < } be the unit disk and T = { z : | z | = 1 } be theunit circle. A bounded analytic function ϑ in D is said to be inner if itsnon-tangential boundary values satisfy | ϑ | = 1 almost everywhere on T . Wedenote by H ( D ) the Hardy space on D and by K ϑ = H ( D ) ⊖ ϑH ( D ) thecorresponding model space.In this article two classes of operators are considered: embeddings I µ : K ϑ → L ( µ ), where µ is a finite positive measure supported on D , andcomposition operators C ϕ : f f ◦ ϕ acting from K ϑ into H ( D ), where ϕ : D → D is a holomorphic function. In fact, it is well-known that thelatter type of operator may be considered as a special case of the former fora certain pullback measure µ ϕ . We mention that embeddings of model spaceshave been studied by a number of authors [4–7, 28]; composition operatorson Hardy (and more general) spaces is by now a classical subject – we referthe reader to [24] for a description of the current state of the art and ahistory survey. In this article we study the composition operator actingfrom the model space K ϑ into H ( D ) thus emphasis interaction betweenthe boundary behavior of ϕ and the spectrum of the inner function ϑ . In Mathematics Subject Classification.
Primary 47B33; Secondary 30H10, 30J05,47A45.
Key words and phrases.
Embedding, composition operator, model space, Nevanlinnacounting function, one-component inner function, Schatten classes.This work was carried out at the Center for Advanced Study, Norwegian Academy ofScience and Letters. Yu.L. and E.M. are partially supported by project 213638 of theNorwegian Research Council. such setting the problem has been considered in [20]. Our main goal is tounderstand when such embedding and composition operators belong to theSchatten trace ideal S p , 0 < p < ∞ .The embedding operators on K ϑ have proved easier to analyze when ϑ is a one-component inner function, see [2, 4, 5, 28]. In particular, the Schat-ten ideal membership of I µ has been characterized by Baranov [4] for one-component ϑ . In Section 3 we suggest a different approach to the problem.We return to the the original ideas of Cohn [6] and extend embedding op-erators on K ϑ to operators acting on the Hardy-Smirnov space E ( D ) of acertain domain D ⊃ D . This allows us to obtain a geometrical criterion forthe inclusion of I µ in S p . In particular we recover the aforementioned resultin [4].For composition operators C ϕ we further refine our result to give traceideal criteria in terms of the Nevanlinna counting function N ϕ , N ϕ ( z ) = X ϕ ( ζ )= z log 1 | ζ | . We combine the geometric approach with recent results [16, 17] that clar-ify the connection between the Nevanlinna counting function N ϕ and themeasure µ ϕ , in order to obtain the following characterization. Theorem 4.2.
Let ϑ be a one-component inner function. The operator C ϕ : K ϑ → H is in S p , < p < ∞ , if and only if Z D (cid:18) N ϕ ( z )(1 − | ϑ ( z ) | ) − | z | (cid:19) p/ (cid:18) − | ϑ ( z ) | − | z | (cid:19) dA < ∞ . The article is organized as follows. The next section contains preliminaryinformation about one-component inner functions and the correspondingmodel spaces. In Section 3 we reduce the trace ideal problem of embeddingoperators on K ϑ to a corresponding problem in the Hardy-Smirnov space ina larger domain, leading to a geometric characterization. In Section 4 we usethese results in order to describe when C ϕ ∈ S p in terms of the Nevanlinnacounting function of ϕ . We also give some geometric examples, illustratingthe Schatten class behavior of composition operators on the Paley-Wienerspace. General model spaces are treated in Section 5, giving an examplethat the one-component requirement of Theorem 4.2 can not be dropped,and providing sufficient conditions for C ϕ to belong to S p , p ≥ Preliminaries
One-component inner functions.
We recall that the class of one-component inner functions, introduced in [6], consists of those inner functions ϑ such that, for some q ∈ (0 , D ǫ = { z ∈ D : | ϑ ( z ) | < ǫ } RACE IDEAL CRITERIA ON MODEL SPACES 3 is connected for all ǫ ∈ ( q , ϑ and a corresponding number q .Consider the canonical factorization of ϑ , ϑ ( z ) = B Λ ( z ) exp (cid:18) − Z T ξ + zξ − z dω ( ξ ) (cid:19) , where Λ is the zero set of ϑ , B Λ is the corresponding Blaschke product, and ω is a singular measure on T . Functions in K ϑ admit analytic continuationthrough T \ Σ( ϑ ), whereΣ( ϑ ) = ( T ∩ Clos(Λ)) ∪ supp( ω )is the spectrum of ϑ (see [22], Lecture 3). The function ϑ itself can bereflected over T \ Σ( ϑ ) and thus definition of the domain D ǫ makes sense forany ǫ ∈ ( q , q − ).Our construction is based on the following results from [6]. Proposition 2.1.
Let δ ∈ (1 , q − ) . Then • The set D δ is simply connected, its boundary ∂D δ is a rectifiableJordan curve, and the linear measure on ∂D δ is a Carleson measurewith respect to D − = { z : | z | > } . • Any function f ∈ K ϑ extends analytically to a function in E ( D δ ) and k f k K ϑ ≃ k f k E ( D δ ) , f ∈ K ϑ . Here and in what follows E ( D δ ) and E − ( D δ ) denote the Hardy-Smirnovspaces in the interior and exterior of D δ (see e.g. [10]). We mention thatfunctions g ∈ E − ( D δ ) are required to assume the value 0 at ∞ .Recall that a rectifiable curve Γ is called Ahlfors regular if for each z ∈ C and r > H (Γ ∩ B ( z, r )) < Cr , where H ( · ) denotes arc length, B ( z, r ) = { ζ : | ζ − z | < r } , and C = C (Γ) > Lemma 1.
Let δ ∈ (1 , q − ) . Then the boundary of D δ satisfies the Ahlforsregularity condition.Proof. Denote q = δ − . The boundary of D δ is a rectifiable Jordan curvethat is the reflection of the curve ∂D q = ϑ − ( | z | = q ) ∪ Σ( ϑ ) with respectto the unit circle, see [6].First, if B ( z, r ) is a ball centered at a point z ∈ Σ( ϑ ), r > C > S (˜ z ) cen-tered at ˜ z = (1 − Cr ) z contains B ( z, r ) ∩ ∂D q . Since the arc length on ∂D q is a Carleson measure, we obtain H ( B ( z, r ) ∩ ∂D q ) ≤ H ( S (˜ z ) ∩ ∂D q ) . − | ˜ z | ≤ Cr, which is precisely the Ahlfors regularity estimate for points z ∈ Σ( ϑ ).For points in the level set Γ q = ϑ − ( | z | = q ), we again want to show theexistence of constant C , independent of z , such that H (Γ q ∩ B ( z, r )) ≤ Cr , A. ALEMAN, YU. LYUBARSKII, E. MALINNIKOVA, K.-M. PERFEKT for all z ∈ Γ q . By the same argument as in the previous paragraph, for afixed a > r > a (1 − | z | ).For comparatively smaller balls, we note that Theorem 1.1 and Lemma2.1 of [6] imply that there exists γ = γ ( q ) such that for any z ∈ Γ q therestriction ϑ | B ( z, γ (1 −| z | )) is univalent. The proof is then completed by theHayman-Wu theorem [14], since it implies that H ( ϑ − ( | z | = q ) ∩ B ( z, r )) . r whenever r < γ (1 − | z | ). (cid:3) Corollary 2.2.
The space L ( ∂D δ ) admits the direct sum decomposition L ( ∂D δ ) = E ( D δ ) ∔ E − ( D δ ) . The corresponding projectors P ± are bounded and have the form (1) P ± f ( z ) = ± f ( z ) + 12 iπ Z ∂D δ f ( ζ ) ζ − z dζ, z ∈ ∂D δ . The proof is straightforward; it suffices to mention that the Ahlfors reg-ularity yields the boundedness of P ± in L ( ∂D δ ) by David’s theorem [9].2.2. Model spaces.
Each function f ∈ K ϑ = H ( D ) ∩ ϑH − ( D ) admits anextension to a function in E ( D δ ). Denote by J the operator of analyticprolongation from D to D δ , and let K ϑ = E ( D δ ) ∩ ϑE − ( D δ ). Proposition 2.3. J ( K ϑ ) = K ϑ .Proof. The inclusion J ( K ϑ ) ⊂ K ϑ follows from Cohn’s extension construc-tion [6] which shows that J ( K ϑ ) ⊂ E ( D δ ), and the observation that if f ∈ K ϑ then ϑ − f has an analytic continuation in C \ D .In order to prove the opposite inclusion we first observe that the linearmeasure ds = | dz | on T is a Carleson measure for E ( D δ ). In other words Z T | f ( z ) | ds . k f k E ( D δ ) , f ∈ E ( D δ ) . Indeed, let f ∈ E ( D δ ). It is sufficient to bound R T f h ds for h ∈ L ( T )that are compactly supported in T \ Σ( ϑ ). For such h we have, by Cauchy’sformula Z T f ( z ) h ( z ) ds = Z ∂D δ f ( ζ ) Z T h ( z ) ζ − z ds ( z ) dζ. The function H ( ζ ) = R T h ( z ) ζ − z ds ( z ) belongs to H − ( D ). Since the arc lengthmeasure on ∂D δ is a Carleson measure for H − ( D ) we have k H k L ( ∂D δ ) . k H k H − ( D ) . k h k L ( T ) and therefore (cid:12)(cid:12)(cid:12)(cid:12)Z T f ( z ) h ( z ) ds (cid:12)(cid:12)(cid:12)(cid:12) . k f k E ( ∂D δ ) k h k L ( T ) , which is the required estimate.The proposition now follows readily. Indeed, let f ∈ K ϑ = E ( D δ ) ∩ ϑE − ( D δ ), so that f | D ∈ H ( D ) and f = ϑg , where g ∈ E − ( D δ ). Since f RACE IDEAL CRITERIA ON MODEL SPACES 5 and ϑ are holomorphic and 1 ≤ | ϑ ( z ) | ≤ δ in D δ \ D , we see that g admitsprolongation to a function in H − ( D ). Hence f | D ∈ H ( D ) ∩ ϑH − ( D ) = K ϑ . (cid:3) Operator extension
Embeddings of E ( D δ ) . We have seen that K ϑ can be considered asa subspace of E ( D δ ). The following theorem reduces trace ideal problemsfor embeddings of K ϑ to trace ideal problems for embeddings of the wholespace E ( D δ ). For a positive measure µ , we denote by I µ the embeddingoperator (inclusion map) from a space into L ( µ ). Theorem 3.1.
Let ϑ be a one-component inner function, µ be a positivemeasure on D \ Σ( ϑ ) , and < p < ∞ . Then the embedding I µ : K ϑ → L ( µ ) is bounded, compact, or belongs to S p if and only if I µ : E ( D δ ) → L ( µ ) isbounded, compact, or belongs to S p , respectively.Proof. We focus on showing that I µ : K ϑ → L ( µ ) belongs to S p if and onlyif I µ : E ( D δ ) → L ( µ ) does. The statements concerning boundedness andcompactness have very similar, but slightly easier proofs. We may furtherassume that dist(Σ( ϑ ) , supp µ ) >
0, as one can easily see that all estimatesare uniform with respect to this quantity.By Proposition 2.3, I µ : K ϑ → L ( µ ) belongs to S p if and only if I µ : K ϑ → L ( µ ) does. From Corollary 2.2 follows the decomposition E ( D δ ) = K ϑ ∔ ϑE ( D δ ) , with bounded projectors P = M ϑ P − M ϑ − and Q = M ϑ P + M ϑ − onto K ϑ and ϑE ( D δ ), respectively. Here P ± are defined in (1), and M h is themultiplication operator with symbol h . We identify the functions in E ( D δ )with their boundary values and consider by extension P and Q as operatorson L ( ∂D δ ). Let C = max( k P + k , k P − k ). Since | ϑ | = δ a.e. on ∂D δ , we have k P k , k Q k ≤ C and also k P + M ϑ − k ≤ a := Cδ − . We estimate the singularvalue of the second summand in the decomposition(2) I µ = I µ P + I µ Q Introducing ˜ ϑ ( z ) = ϑ ( z ) | supp µ , we have I µ Q = M ˜ ϑ I µ P + M ϑ − . Weobtain the following estimate of the j ’th singular value of I µ Q : s j ( I µ Q ) ≤ k M ˜ ϑ k s j ( I µ ) k P + M ϑ − k ≤ as j ( I µ ) . Here we used the fact that | ˜ ϑ ( z ) | ≤ z ∈ supp µ .Combining (2) with known (see e.g. [12]) inequalities for singular numberswe obtain n X s j ( I µ ) ≤ n X s j ( I µ P ) + n X s j ( I µ Q ) ≤ n X s j ( I µ P ) + a n X s j ( I µ ) . If a < p ≥ k I µ : E ( D δ ) → L ( µ ) k S p ∼ k I µ : K ϑ → L ( µ ) k S p . A. ALEMAN, YU. LYUBARSKII, E. MALINNIKOVA, K.-M. PERFEKT
More generally, such an equivalence of ideal norms holds for any sym-metrically normed ideal of compact operators [12]. For p < s j − ( I µ ) ≤ s j ( I µ P ) + as j ( I µ ). If a < a is small enough, where 2 p +1 a p = 1,we see that X j s j ( I µ P ) p ∼ X j s j ( I µ ) p , finishing the proof also in this case.To deal with general values of a , we note that I µ : K ϑ → L ( µ ) is in S p if and only if I µ : K ϑ → L ( µ ) is in S p , since K ϑ ⊂ K ϑ = K ϑ ⊕ ϑK ϑ .Replacing ϑ by a sufficiently large power ϑ n we will obtain a new value a = Cδ − n such that k P + M ϑ − n k ≤ a < a <
1; note that in moving fromthe study of K ϑ to that of K ϑ n we do not change the domain D δ , so thatthe projections P ± stay the same for all values of n . (cid:3) Whitney decomposition.
To pass from the domain D δ to the unitdisk, let σ be a conformal mapping of D onto D δ , and ψ be its inverse. For f ∈ E ( D δ ), let h f ( w ) = f ( σ ( w ))( σ ′ ( w )) / , w ∈ D . Let further ν denotethe measure on D given by ν ( E ) = Z σ ( E ) | ψ ′ | dµ, E ⊂ D . Then for f, g ∈ E ( D δ ) we have( I ∗ µ I µ f, g ) = Z D \ Σ( ϑ ) f ( z ) g ( z ) dµ ( z ) = Z D h f ( w ) h g ( w ) dν ( w ) . Therefore, by Theorem 3.1, the embedding I µ : K ϑ → L ( µ ) is in S p if andonly if the embedding I ν : H ( D ) → L ( ν ) is in S p . We can now applythe results from Luecking [18] to conclude that I µ : K ϑ → L ( µ ) is in theSchatten ideal S p , 0 < p < ∞ , if and only if(3) X j (cid:18) ν ( R j ) d ( R j ) (cid:19) p/ < ∞ , where { R j } is the standard dyadic decomposition of the unit disk, and d ( R j ) = diam( R j ).For a domain Ω in the plane, we say that a family of Borel sets { G i } i is a Whitney-type decomposition of Ω if Ω = ∪ i G i , the covering is of finitemultiplicity, and there exist constants a, b, c such that: (i) if z , z ∈ G i thendist( z , ∂ Ω) ≤ c dist( z , ∂ Ω), (ii) for each i there exists z ∈ G i such that B ( z, ad ) ⊂ G i ⊂ B ( z, bd ), where d = dist( z, ∂ Ω).We need the following simple observation:
Given two Whitney-type decompositions { G i } i and { F j } j of a domain Ω ,let J ( i ) = { j, G i ∩ F j = ∅} . Then M := sup i | J ( i ) | < ∞ . Indeed, just note that all F j that intersect G i have diameter proportionalto d i = dist( G i , ∂ Ω) and the area of each such F j is proportional to d i . RACE IDEAL CRITERIA ON MODEL SPACES 7
Furthermore, if φ : Ω → C is univalent, then { φ ( G i ) } i is a Whitney-typedecomposition of φ (Ω). This follows from standard estimates for univalentfunctions, see e.g. [23]. Together with Luecking’s condition (3) this yieldsthe following corollary. Corollary 3.2.
Let ϑ be a one-component inner function, D δ be a corre-sponding level set and µ be a positive measure on D \ Σ( ϑ ) . Further, let { G i } be any Whitney-type decomposition of D δ . Then the embedding K ϑ → L ( µ ) belongs to the Schatten ideal S p , < p < ∞ , if and only if (4) X i (cid:18) µ ( G i ) d ( G i ) (cid:19) p/ < ∞ , where d ( G i ) = diam( G i ) .Proof. First let Q j = σ ( R j ), where σ : D → D δ is a conformal mapping,as above. Then { Q j } j is a Whitney-type decomposition of D δ and since | σ ′ | ∼ dist( Q j , ∂D δ ) / dist( R j , ∂ D ) ∼ diam( Q j ) / diam R j (see [23]), (3) isequivalent to X i (cid:18) µ ( Q j ) d ( Q j ) (cid:19) p/ < ∞ . But for any α > X i (cid:18) µ ( G i ) d ( G i ) (cid:19) α ∼ X i,j µ ( G i ∩ Q j ) α d ( G i ) α ∼ X i,j µ ( G i ∩ Q j ) α d ( Q j ) α ∼ X i (cid:18) µ ( Q j ) d ( Q j ) (cid:19) α , proving the corollary. (cid:3) We remark that Corollary 3.2, which will be the main ingredient in theproof of Theorem 4.2 below, can be deduced from a result of Baranov [4]. Wethink however that Theorem 3.1 may be of independent interest and it hassome applications which do not appear to us to be immediate consequencesof Corollary 3.2, see Section 4.3.The proof of Corollary 3.2 also gives a simple and natural criterion forthe Schatten class memberships of embeddings of the Hardy-Smirnov space E ( D ) into L ( µ ) for measures µ in D ; one has to check the Luecking con-dition for an arbitrary Whitney-type decomposition of D .4. Composition operators
Preliminaries.
For a holomorphic function ϕ : D → D , we denote by C ϕ : f f ◦ ϕ the composition operator acting on holomorphic functions f in D . This operator is bounded on the Hardy space H ( D ) (see e.g. [26]).We study the operator C ϕ : K ϑ → H ( D ), where ϑ is an inner function in D . The compactness of C ϕ in terms of the Nevanlinna counting function N ϕ ( z ) = X ϕ ( ζ )= z log 1 | ζ | A. ALEMAN, YU. LYUBARSKII, E. MALINNIKOVA, K.-M. PERFEKT was characterized in [20]; C ϕ : K ϑ → H is compact if and only if(5) lim sup | z |→ N ϕ ( z )(1 − | ϑ ( z ) | )1 − | z | = 0 . The basic tools in the argument are the Stanton formula(6) k C ϕ f k = 2 Z D | f ′ ( z ) | N ϕ ( z ) dA ( z ) + | f ( ϕ (0)) | , where A is the normalized area measure, and also the norm inequality dueto Axler, Chang and Sarason [3](7) Z D | f ′ ( z ) | − | z | (1 − | ϑ ( z ) | ) b ≤ C k f k , f ∈ K ϑ , b ∈ (0 , / . In this section we discuss when C ϕ belongs to the Schatten ideals S p in the one-component case, aiming to capture the interaction between thesymbol ϕ and the inner function ϑ that defines the model space. We recallthe known description of the Schatten ideals for composition operators onthe whole of H , due to Luecking and Zhu [19]. The operator C ϕ belongsto S p ( H ) if and only if(8) Z D (cid:18) N ϕ ( z )1 − | z | (cid:19) p/ dA ( z )(1 − | z | ) < ∞ . It is well understood that the composition operators can be consideredas a special case of the embedding operators, see e.g. [8]. We shall nowclarify this connection in our context, so that we may apply Theorem 3.1and Corollary 3.2. We associate ϕ : D → D with its pullback measure µ ϕ on D ; µ ϕ ( E ) = s ( ϕ − ( E ) ∩ T ) , E ⊂ D , where s denotes the Lebesgue measure on T .It is clear that C ϕ : K ϑ → H is unitarily equivalent to the embeddingoperator I µ ϕ : K ϑ → L ( µ ϕ ), and similarly that C ϕ : E ( D δ ) → H isequivalent to I µ ϕ : E ( D δ ) → L ( µ ϕ ). Before applying Theorem 3.1 weneed to verify that µ ϕ (Σ( ϑ )) = 0. This is true in view of the followinglemma and the fact that Σ( ϑ ) has zero linear Lebesgue measure when ϑ isone-component, see [2]. Lemma 2. µ ϕ | T is absolutely continuous.Proof. It is sufficient to verify that µ ϕ ( E ) = 0 for every closed measure zeroset E ⊂ T . We follow the approach of the original proof of the F. andM. Riesz theorem. Namely, there exists a continuous function G : D → D ,holomorphic in D , such that G ( z ) = 1 for z ∈ E and | G ( z ) | < z ∈ D \ E .Then lim k R D G k dµ ϕ = µ ϕ ( E ). On the other hand, the sequence G k ◦ ϕ converges pointwise to zero in D and is uniformly bounded. Therefore, µ ϕ ( E ) = lim k Z D G k dµ ϕ = lim k Z T G k ◦ ϕ ds = lim k ( G ( ϕ (0))) k = 0 . RACE IDEAL CRITERIA ON MODEL SPACES 9 (cid:3)
Theorem 3.1 now yields
Corollary 4.1.
Let ϑ be a one-component inner function and ϕ : D → D be a holomorphic function. Given any Whitney-type decomposition { G j } of D δ , the operator C ϕ : K ϑ → H belongs to S p , < p < ∞ , if and only if (9) X i (cid:18) µ ϕ ( G i ) d ( G i ) (cid:19) p/ < ∞ . Nevanlinna counting function.
In this subsection we implement theapproach of [16, 17] in our more general setting, with the goal of showingthat (9) is equivalent to Z D δ (cid:18) N ϕ ( z )dist( z, ∂D δ ) (cid:19) p/ dA ( z )dist( z, ∂D δ ) < ∞ . Theorem 4.2 then immediately follows from the relation(10) dist( z, ∂D δ ) ∼ − | z | − | ϑ ( z ) | , z ∈ D \ Σ( ϑ ) , which holds for one-component inner functions, by Theorems 1.1 and 1.2 inAleksandrov [2].We begin by fixing a convenient Whitney-type decomposition of D δ .Clearly we are interested only in domains that intersect D . We constructa decomposition of A = { / < | z | ≤ } \ Σ( ϑ ) as follows. First we di-vide A into four equal parts, one for each quadrant. Each part is roughlya Carleson square. Fix some γ >
0. We say that a Carleson square S is good if dist( S, ∂D δ ) > γd ( S ). If a square is good we include it in our fam-ily of sets { G i } . Otherwise we include its upper half into the family { G i } and divide the lower half into two new Carleson squares. We repeat theprocedure inductively, obtaining a countable family of sets { G i } that covers A . In particular, every point z ∈ T \ Σ( ϑ ) is included in a good square,since Σ( ϑ ) is a closed set. We claim that dist( G i , ∂D δ ) ≃ d ( G i ) for each G i . For if G i is the upper half of a bad square S , it automatically satisfies d ( G i ) . dist( G i , ∂D δ ). Since S was bad we obtain the reverse inequality,dist( G i , ∂D δ ) . d ( G i ) + dist( S, ∂D δ ) ≤ d ( G i ) + γd ( S ) . d ( G i ) . A similar argument works for the good squares G i . Further, { G i } can beextended to a Whitney-type decomposition of the whole D δ , since supp µ ⊂ D . We omit the corresponding terms.For each G i we let W i be the corresponding Carleson square (i.e. either W i = G i or G i is the upper half of W i ). Given a Carleson square W supported by the arc I ⊂ T and a > aW the Carlesonsquare supported by the arc aI ⊂ T . aI has the same center as I and | aI | = a | I | . According to [16] and [17] there exists a > c > µ ( W i ) p/ ≤ cA ( W i ) Z ˜ W i N p/ ϕ dA, (12) sup W i N ϕ ≤ cµ ( ˜ W i ) , where ˜ W i = aW i .To obtain the integral condition, we follow the argument of [15, Proposi-tion 3.3] and prove that (9) for the above decomposition { G i } i is equivalentto(13) X i µ ϕ ( ˜ W i ) d ( W i ) ! p/ < ∞ . Clearly (13) implies (9). Now we prove the converse statement.For each j and n let I ( j ) = { i : ˜ W i ∩ G j = ∅} and S n,j = { i ∈ I ( j ) : 2 n d ( G j ) ≤ d ( W i ) < n +1 d ( G j ) } . We note that d ( G j ) ∼ dist( G j , ∂D δ ) . d ( W i ) for any i ∈ I ( j ), yieldingthat S n,j = ∅ when n < n for some fixed (negative) n . We also havedist( G j , W i ) . d ( W i ) which yields C := sup j,n | S n,j | < ∞ .For α := p/ ≤ X i µ ϕ ( ˜ W i ) d ( W i ) ! α ≤ X i X j : i ∈ I ( j ) µ ϕ ( G j ) α d ( W i ) α = X j µ ϕ ( G j ) α X i ∈ I ( j ) d ( W i ) − α . X j (cid:18) µ ϕ ( G j ) d ( G j ) (cid:19) α , as desired.For α > i , considerthe sets s n,i = { j : i ∈ I ( j ) , − n − d ( W i ) ≤ d ( G j ) < − n d ( W i ) } . We have | s n,i | ≤ C n for some constant C . Let β be the conjugate exponentof α and choose γ ∈ (1 − /α,
1) such that 1 − γβ <
0. By the H¨olderinequality µ ϕ ( ˜ W i ) ≤ X j : i ∈ I ( j ) d ( G j ) γβ /β X j : i ∈ I ( j ) d ( G j ) − γα µ ϕ ( G j ) α /α . d ( W i ) γ X j : i ∈ I ( j ) d ( G j ) − γα µ ϕ ( G j ) α /α , RACE IDEAL CRITERIA ON MODEL SPACES 11 so that finally, X i µ ϕ ( ˜ W i ) d ( W i ) ! α ≤ X j µ ϕ ( G j ) α d ( G j ) − γα X i ∈ I ( j ) d ( W i ) γα − α . X j (cid:18) µ ϕ ( G j ) d ( G j ) (cid:19) α . The sought after criterion in terms of the Nevanlinna counting functionnow follows readily.
Theorem 4.2.
Let ϑ be a one-component inner function. The operator C ϕ : K ϑ → H is in S p , < p < ∞ , if and only if Z D (cid:18) N ϕ ( z )(1 − | ϑ ( z ) | ) − | z | (cid:19) p/ (cid:18) − | ϑ ( z ) | − | z | (cid:19) dA < ∞ . Proof.
We follow the proof for the Hardy space given in [17, Theorem 6.1].Inequality (12) for the Nevanlinna counting function implies Z D (cid:18) N ϕ ( z )dist( z, ∂D δ ) (cid:19) p/ dA ( z )dist( z, ∂D δ ) . X i d ( G i ) − − p/ Z G i N ϕ ( z ) p/ dA ( z ) . X i d ( G i ) − p/ µ ϕ ( ˜ W i ) p/ . The converse follows from inequality (11). Specifically, X i d ( G i ) − p/ µ ϕ ( G i ) p/ ≤ X i d ( G i ) − p/ − Z ˜ W i N ϕ ( z ) p/ dA ( z ) . X j X i ∈ I ( j ) d ( G i ) − p/ − Z G j N ϕ ( z ) p/ dA ( z ) . Z D (cid:18) N ϕ ( z )dist( z, ∂D δ ) (cid:19) p/ dA ( z )dist( z, ∂D δ ) , where the last inequality follows as in the discussion preceding the statementof the theorem. (cid:3) Weighted composition operators on H . We complete the study ofcomposition operators on model spaces generated by one-component innerfunctions, by establishing the connection given by Theorem 3.1 betweencomposition operators on K ϑ and weighted composition operators on H ( D ).As previously noted, C ϕ : K ϑ → H is of S p -class if and only if C ϕ : E ( D δ ) → H has the same property. That is, if and only if the weightedcomposition operator C : H → H is of S p -class, where Ch = ( ψ ′ ◦ ϕ ) / h ◦ ψ ◦ ϕ, h ∈ H . We mention [11, 21], where composition operators on Hardy-Smirnov spaces E ( D ) have been studied as weighted composition operators on H ( D ). Notethat Ch = h ( M ψ ◦ ϕ )( ψ ′ ◦ ϕ ) / , at least for polynomials h . Here M ψ ◦ ϕ denotesa multiplication operator on H , and we hence understand Ch as the action of the operator h ( M ψ ◦ ϕ ) on ( ψ ′ ◦ ϕ ) / ∈ H . For p ≥
1, Harper andSmith [13] have utilized the theory of contractive semigroups to characterizethe Schatten membership of such operators in terms of Berezin transform-type conditions. Note that in their notation, C = Λ M ψ ◦ ϕ , ( ψ ′ ◦ ϕ ) / . Theorem 4.3.
Denote by G z ( w ) = (1 −| z | ) / − ¯ zw the normalized reproducingkernel of H at z and by H z ( w ) = (1 −| z | ) / w (1 − ¯ zw ) the normalized derivative ofthe reproducing kernel at z . Let ϑ be a one-component inner function andlet C be defined as above. (1) If ≤ p ≤ , then C ϕ : K ϑ → H belongs to S p if and only if Z D k CH z k pH dA ( z )(1 − | z | ) < ∞ . (2) If < p < ∞ , then C ϕ : K ϑ → H belongs to S p if and only if Z D k CG z k pH dA ( z )(1 − | z | ) < ∞ . Remark.
For ϑ = 0, p ≥
2, we obtain the characterization of the Schattenclasses in terms of the Berezin transforms found in [29]. For p = 2, notethat Z D k CH z k H dA ( z )(1 − | z | ) = Z D | ψ ′ ( w ) || ψ ( w ) | Z D − | z | | − ¯ zψ ( w ) | dA ( z ) dµ ϕ ( w ) ∼ Z D | ψ ′ ( w ) || ψ ( w ) | − | ψ ( w ) | dµ ϕ ( w ) . Note that 1 − | ψ ( w ) | ∼ dist( w, ∂D δ ) | ψ ′ ( w ) | by standard estimates [23], andthat we may reexpress (10) asdist( w, ∂D δ ) ∼ − | w | − | ϑ ( w ) | = k k w k − , w ∈ D \ Σ( ϑ ) , where k w is the reproducing kernel of K ϑ at w . In summary, Z D k CH z k H dA ( z )(1 − | z | ) ∼ Z D k k w k dµ ϕ ( w ) . Theorem 4.3 could hence be viewed as a generalization of the simple factthat C ϕ : K ϑ → H is Hilbert-Schmidt if and only if R D k k w k dµ ϕ < ∞ .4.4. Examples on the Paley–Wiener space.
In this section we considerthe special case ϑ ( z ) = exp (cid:16) − z − z (cid:17) . The space K ϑ can then be naturallyidentified with the classical Paley–Wiener space of entire functions.For 0 < α < U α ⊂ D satisfying the followingproposition. The domain U α will be chosen so that the boundary ∂U α in-tersects T only at 1 and has a corner of angle πα there. Proposition 4.4.
For each α , < α < , there exists a domain U α suchthat any corresponding Riemann map ϕ α : D → U α satisfies RACE IDEAL CRITERIA ON MODEL SPACES 13 (1) C ϕ α : H → H is compact but not in any Schatten class S p , (2) C ϕ α : K ϑ → H is in S p for p > α/ ( α − . Note that C ϕ α : K ϑ → H is in S p for one Riemann map ϕ α : D → U α if and only if it is in S p for all such Riemann maps. The same statement isobviously true for C ϕ α : H → H .Let V α ⊂ D be the simply connected domain whose boundary consistsof the upper half-circle { z : | z | = 1 , ℑ z ≥ } and the circular arc withterminal points − πα at those points. Note that V α = κ ( A α ), where A α = { re iθ : 0 < r < ∞ , < θ < πα } , and κ is the M¨obius map κ ( w ) = 1 + iw − iw . Let ψ α : D → V α be a Riemann map. The next lemma says that C ψ α : K ϑ → H is in S p if and only if p > α/ ( α − U α as a subset of V α , so that C ϕ α = C τ C ψ α for a composition operator C τ : H → H with τ a Riemannmap from D to ψ − α ( U α ). Lemma 3. C ψ α : K ϑ → H is p -Schatten if and only if p > α/ ( α − .Proof. Y. Zhu [30] obtains the corresponding result on H , for domainssimilar to V α , rescaled and translated to only touch (tangentially from oneside) the unit circle at z = 1. In view of the fact that C ψ α : K ϑ → H belongs to S p if and only if C ψ α : E ( D δ ) → H belongs to S p and theobservation that ∂D δ in this case is a circle such that T is internally tangentto ∂D δ at 1, the lemma follows. A direct computational proof based on ourintegral criterion can be also given. (cid:3) We now define U α . Let γ α be the circular arc of ∂V α which is not theupper half-circle, γ α = { z ∈ ∂V α : | z | 6 = 1 } ∪ {− , } . Close to 1, we choose ∂U α to coincide with the union of γ α and τ = { re iθ : 0 < θ ≤ π/ , r = 1 − e − /θ } . That is, let U α be a simply connected domain contained in V α , such that ∂U α ∩ T = { } and for sufficiently small ǫ > ∂U α ∩ B ǫ (1) = ( γ α ∪ τ ) ∩ B ǫ (1) , where B ǫ (1) is a ball centered at 1 of radius ǫ . Proof of Proposition 4.4.
We have already proven the validity of item (2).It remains to prove that C ϕ α : H → H is compact but fails to lie inany Schatten class S p . For simplicity in presentation we will assume that α < /
2. The proof for α ≥ / ϕ α (1) = 1. Let η α = ϕ − α . For z ∈ U α close to 1 we have that N ϕ α ( z ) ∼ − | η α ( z ) | .We will again make use of the inequality1 − | η α ( z ) | ∼ dist( z, ∂U α ) | η ′ α ( z ) | . Since η α maps U α , a domain with a corner of angle πα at 1, conformallyonto the unit disc, it follows that | η ′ α ( z ) | ∼ | z − | /α − for z close to 1 (seee.g. [23]). Therefore, for z ∈ U α close to 1 we have N ϕ α ( z ) ∼ dist( z, ∂U α ) | z − | /α − . This estimate obviously implies that lim | z |→ N ϕα ( z )1 −| z | = 0 so that C ϕ α iscompact on H [26], and we will now use it to show that Z U α ∩ B ǫ (1) N ψ α ( z ) p/ (1 − | z | ) p/ dA ( z ) = ∞ for every p >
0, proving in view of (8) the desired statement.We will restrict ourselves to considering points in the set U ′ α = { z = re iθ ∈ B ǫ (1) : r < − e − /θ , θ > k (1 − r ) } , for a fixed large constant k . k should be chosen so large and then ǫ so smallthat there exists a constant C > / tan( πα ) such that every point re iθ ∈ U ′ α satisfies r sin θ > C (1 − r cos θ ). In other words, U ′ α should lie above the line ℑ z = C (1 − ℜ z ), which intersects the unit circle in the point 1 at an anglesmaller than πα . If necessary, we decrease ǫ further, attaining that U ′ α ⊂ U α and dist( z, ∂U α ) ∼ − e − /θ − r ∼ − | z | , z ∈ U ′ α . The proof is now finished by the following chain of inequalities. Z U ′ α N ψ α ( z ) p/ (1 − | z | ) p/ dA ( z ) ∼ Z U ′ α | − z | p (1 − α )2 α (1 − | z | ) dA ( z ) & Z − ǫ ′ Z − r k (1 − r ) (1 − r cos θ + r ) p (1 − α )4 α (1 − r ) dθ dr & Z − ǫ ′ Z − r k (1 − r ) θ p (1 − α )2 α (1 − r ) dθ dr & Z − ǫ ′ (cid:18) − r (cid:19) p (1 − α )2 α +1 (1 − r ) dθ dr = ∞ , where 0 < ǫ ′ < ǫ is sufficiently small. (cid:3) Remarks on composition operators on general model spaces
In this section we do not assume that the inner function ϑ is one-component. RACE IDEAL CRITERIA ON MODEL SPACES 15
The Hilbert-Schmidt norm.
Denote the reproducing kernel of K ϑ at w by k ( w, · ), k ( w, z ) = 1 − ϑ ( w ) ϑ ( z )1 − ¯ wz , z ∈ D . Proposition 5.1.
The Hilbert-Schmidt norm of the composition operator C ϕ : K ϑ → H is given by the expression k C ϕ k HS = 1 − | ϑ (0) | + 12 Z D ∆ k ( z, z ) N ϕ ( z ) dA ( z ) . Proof.
Let { e j ( z ) } be any orthonormal basis in K ϑ . Then k ( z, z ) = P j | e j ( z ) | and the Stanton formula (6) yields k C ϕ k HS = X j ( C ϕ e j , C ϕ e j ) = X j (cid:18) | e j (0) | + 2 Z D | e ′ j | N ϕ dA (cid:19) . Since P j | e ′ j ( z ) | = ∆ P j | e j ( z ) | , we obtain k C ϕ k HS = k (0 ,
0) + 12 Z D ∆ k ( z, z ) N ϕ ( z ) dA ( z ) . (cid:3) Corollary 5.2. (i) If Z D − | ϑ ( z ) | (1 − | z | ) N ϕ ( z ) dA ( z ) < ∞ then C ϕ : K ϑ → H is a Hilbert-Schmidt operator;(ii) If C ϕ : K ϑ → H is a Hilbert-Schmidt operator then Z D (1 − | ϑ ( z ) | ) (1 − | z | ) N ϕ ( z ) dA ( z ) < ∞ . Proof.
A direct calculation shows that14 ∆ k ( z, z ) = (1 + | z | )(1 − | ϑ ( z ) | )(1 − | z | ) − ℜ ( zϑ ( z ) ϑ ′ ( z ))(1 − | z | ) − | ϑ ′ ( z ) | − | z | . Applying the standard inequality | ϑ ′ ( z ) | ≤ (1 − | ϑ ( z ) | )(1 − | z | ) − ,(14) 4 ( | z | − | ϑ ( z ) | ) (1 − | ϑ ( z ) | )(1 − | z | ) ≤ ∆ k ( z, z ) ≤ | z | ) (1 − | ϑ ( z ) | )(1 − | z | ) . Statement (i) is an immediate consequence of the right hand side inequal-ity in (14).In order to prove statement (ii) we first observe that if C ϕ : K ϑ → H is aHilbert-Schmidt operator then so is C ϕ : K zϑ → H , since K zϑ = K z ⊕ zK ϑ and dim K z = 1. Denote by ˜ k ( w, z ) the reproducing kernel of K zϑ at w . Wethen have Z D \ D (1 − | ϑ ( z ) | ) (1 − | z | ) N ϕ ( z ) ≤ Z D (1 − | zϑ ( z ) | )( | z | − | zϑ ( z ) | ) (1 − | z | ) N ϕ ( z ) dA ( z ) ≤ Z D ∆˜ k ( z, z ) N ϕ ( z ) dA ( z ) ≤ k C ϕ : K zϑ → H k HS . (cid:3) Remark.
In the previous section we proved that the inequality(15) Z D (1 − | ϑ ( z ) | ) (1 − | z | ) N ϕ ( z ) dA ( z ) < ∞ gives a complete description of the Hilbert-Schmidt composition operators C ϕ : K ϑ → H in the case of one-component ϑ . This condition is notsufficient in general, see Section 5.3 below.5.2. A sufficient condition for larger Schatten ideals.
The Hilbert-Schmidt norm characterization, discussed for composition operators so far,remains true for the differentiation operator acting between K ϑ and L ( D , µ ),where µ is a positive, finite measure on D . Using complex interpolation wecan obtain a sufficient condition for a composition operator to be in theSchatten ideal S p when 2 ≤ p < ∞ .First we assume that Φ( z ) is a positive function on D such that the dif-ferentiation D : K ϑ → L (Φ( z ) dA ) is a bounded operator. It follows from(7) that we can take Φ( z ) = 1 − | z | (1 − | ϑ ( z ) | ) b , with b ∈ (0 , / ϑ is a one-component innerfunction we may choose Φ( z ) = 1 − | z | − | ϑ ( z ) | , see [7, Theorem 1]. Proposition 5.3.
Let Φ( z ) be as above and p ≥ . If Z D (cid:18) N ϕ ( z )Φ( z ) (cid:19) p/ ∆ k ( z, z )Φ( z ) dA ( z ) < ∞ then C ϕ : K ϑ → H belongs to S p .Proof. The proof follows an idea of Luecking [18]; see also [4]. By the Stantonformula (6), we want to prove that the differentiation operator D : K ϑ → L ( N ϕ dA ) belongs to the Schatten ideal S p . It is enough to prove thecorresponding statement where N ϕ has been replaced by N = χ r N ϕ , χ r RACE IDEAL CRITERIA ON MODEL SPACES 17 being the characteristic function of the disk r D , as long as the correspondingSchatten norm is independent of r .Consider the following holomorphic family of compact operators( T ζ f )( z ) = N ( z ) p (1 − ζ ) / Φ( z ) / − p (1 − ζ ) / f ′ ( z ) , where ζ ∈ C , 0 ≤ ℜ ζ ≤
1. We note that T iy : K ϑ → L ( dA ) is boundedby a constant C that depends on Φ only. Further, T iy : K ϑ → L ( dA ) is aHilbert-Schmidt operator with norm (see the proof of Proposition 5.1) k T iy k HS ≤ Z D ∆ K ( z, z ) N ( z ) p/ Φ( z ) − p/ dA ( z ) ≤ C by the assumption of the theorem. Then T − /p ∈ S p and k T − /p k S p ≤ C /p C − /p by Theorem 13.1 in [12]. But T − /p f ( z ) = N ( z ) / f ′ ( z ). (cid:3) Note that for ϑ = 0, Proposition 5.3 gives us the known condition ofLuecking and Zhu [19].5.3. Integral condition insufficient when ϑ is not one-component. Let α n = 2 − n , r n = 1 − n − n , z n = r n e iα n , n = 1 , , . . . , and let ϑ be the Blaschke product ϑ ( z ) = ∞ Y n =1 − z n | z n | z − z n − z n z , z ∈ D . The sequence { z n } is interpolating (for H ∞ ), since it is hyperbolically sep-arated in D and P n (1 − | z n | ) δ z n is a Carleson measure for H . Thereforethe sequence of normalized reproducing kernels { (1 − | z n | ) / k z n } n , k z n = 11 − ¯ z n z , is a Riesz basis for K ϑ . In particular any Hilbert-Schmidt operator C : K ϑ → H satisfies(16) X n (1 − | z n | ) h Ck z n , Ck z n i < ∞ . Let ∆ = { z : | z − | < } and let ϕ ( z ) = (1 + z ) /
2, a conformal mappingof D onto ∆. We will show that (16) with C = C ϕ fails, while(17) Z D (cid:18) − | ϑ ( z ) | − | z | (cid:19) N ϕ ( z ) dA ( z ) < ∞ . Indeed, since N ϕ ( z ) = − log | z − | & dist( z, ∆) for z ∈ ∆, we have(1 − | z n | ) h C ϕ k z n , C ϕ k z n i H & (1 − | z n | ) Z ∆ (cid:12)(cid:12)(cid:12)(cid:12) z − z n (cid:12)(cid:12)(cid:12)(cid:12) − N ϕ ( z ) dA ( z ) & − | z n | dist( z n , ∆) ∼ n . This yields divergence of the series on the right hand side of (16). Therefore C ϕ is not a Hilbert-Schmidt operator.To prove (17) we need an auxiliary statement. Lemma 4.
For any a > , the sum ∞ X n =1 (cid:18) − | z n || − z n z | (cid:19) a is uniformly convergent in D . In particular, it is uniformly bounded.Proof. For z ∈ D and 0 < t <
1, let M ( t, z ) = { n ≥ −| z n || − z n z | > t } and let N ( t, z ) = | M ( t, z ) | be the number of points in M ( t, z ). Pick n and k suchthat n > k ≥ N ( t, z ) − z n , z k ∈ M ( t, z ). Then2 − k | z | . | ( z n − z k ) z | . − | z k | t ≤ − k t , from which | z | t . − k . − N ( t,z ) . That is, there exists a constant C > N ( t, z ) ≤ log C | z | t . In combination with the identity ∞ X n =1 (cid:18) − | z n || − z n z | (cid:19) α = α Z t α − N ( t, z ) dt, the statement of the lemma clearly follows. (cid:3) In order to prove (17) we first observe that for 0 < ǫ < − | ϑ ( z ) | − | z | ≤ X n − | z n | | − z n z n | ≤ X n (cid:18) − | z n | | − z n z | (cid:19) − ǫ ! / X n (1 − | z n | ) ǫ | − z n z | ǫ ! / . Let ∆ ′ = { z : | z − / | < / } . Using Lemma 4 and the fact that N ϕ ( z ) ∼ dist( z, ∆) for z ∈ ∆ \ ∆ ′ , we obtain Z D \ ∆ ′ (cid:18) − | ϑ ( z ) | − | z | (cid:19) N ϕ ( z ) dA ( z ) . X n (1 − | z n | ) ǫ Z ∆ (cid:12)(cid:12)(cid:12)(cid:12) z − z n (cid:12)(cid:12)(cid:12)(cid:12) − − ǫ dist( z, ∆) dA ( z ) ∼ X n (cid:18) − | z n | dist( z n , ∆) (cid:19) ǫ ∼ X n n ǫ < ∞ , which of course implies (17). RACE IDEAL CRITERIA ON MODEL SPACES 19
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