Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.


Introduction
All over the text, Ω will denote an irreducible symmetric cone in R n , and D = R n + iΩ the tube domain over Ω. As in [10] we denote by r the rank of the cone Ω and by ∆ the associated determinant function in R n . We recall that for n ≥ 3, when r = 2, as example of symmetric cones, we have the Lorentz cone Λ n which is defined by Λ n = {(y 1 , · · · , y n ) ∈ R n : y 2 1 − · · · − y 2 n > 0, y 1 > 0}; its associated determinant function is given by the Lorentz form ∆(y) = y 2 1 − · · · − y 2 n . As usual, we denote by H(D) the space of holomorphic functions on D.
Let 1 ≤ p < ∞. For ν ∈ R we denote by L p ν (D) = L p (D, ∆ ν− n r (y)dx dy) the space of functions f satisfying the condition f p,ν = ||f || L p ν (D) := D |f (x + iy)| p ∆ ν− n r (y)dxdy The weighted Bergman space A p ν (D) is the closed subspace of L p ν (D) consisting of holomorphic functions in D. Following [9], this space is not trivial (i.e. A p ν (D) = {0}) only if ν > n r − 1. The orthogonal projection of the Hilbert space L 2 ν (D) onto its closed subspace A 2 ν (D) is called the weighted Bergman projection and denoted P ν . We recall that P ν is given by with K ν (z, w) = c ν ∆ −(ν+ n r ) ((z−w)/i). We recall that K ν is the reproducing kernel of A 2 ν (D) (see [10]). For simplicity, we used the notation dV ν (w) := ∆ ν− n r (v)du dv, where w = u + iv ∈ D.
For µ a positive Borel measure on D, the Toeplitz operator T µ is the operator defined for functions f with compact support by where K ν is the weighted Bergman kernel. Schatten class S p (0 < p ≤ ∞) criteria of the Toeplitz operators have been considered by many authors on some bounded domains of C n (see [1,8,12,19,20] and the references therein). For unbounded domains, Schatten classes have been also characterized in Fock spaces by several authors (see for example [11,14] and the references therein). In [13], we extended these results for 1 ≤ p ≤ ∞ to weighted Bergman spaces of tube domains over symmetric cones. To be more precise, let us introduce more notations.
For δ > 0, we denote by the Bergman ball centered at z with radius δ, d is the Bergman distance on D. For ν > n r − 1 and w ∈ D, the normalized reproducing kernel of A 2 ν (D) at w is given by Let µ be a positive measure on D. The Berezin transform of the measure µ is the functionμ defined on D bỹ The Berezin transform of a function f is defined to be the Berezin transform of the measures dµ(z) = f (z)dV ν (z) (for more on the Berezin transform, see [18]). For z ∈ D and δ ∈ (0, 1), we define the average of the positive measure µ at z byμ The following was obtained in [13].
Our first concern in this note is for the extension of the above result to the range 0 < p < 1. We prove that the equivalences (i)⇔(ii)⇔(ii) still hold for n r −1 ν+ n r < p < 1. This cut-off is due to integrability conditions of the determinant function. The equivalence with the last assertion in the above result still also holds if we restrict to 2 n r −1 ν+ n r < p < 1. The last cut-off point is also due to integrability conditions of the determinant function and one can prove that it is sharp. Our result is then as follows.
Moreover, if p > 2 n r −1 ν+ n r , then the above assertions are equivalent to the following (iv)μ ∈ L p (D, dλ).
The main difficulty in the proof of the above theorem is the implication (i)⇒(ii). The idea in [20] is to replace the measure µ by a measure supported on a disjoint union of Bergman balls, then split the associated Toeplitz operator into its diagonal and off-diagonal part. It is not hard to prove that the Schatten norm of the diagonal part dominates the l p -norm of the sequence {μ δ (ζ j )}. The difficulty is to prove that the latter norm dominates (up to a pretty small constant) the Schatten norm of the off-diganal operator. A part of the techniques in [20] uses the fact that the unit ball is bounded, and so it cannot be used in our setting. We overcome this difficulty by using a technical lemma originally due to D. Békollé and A. Temgoua [7]. Considered even in the unit ball, our contribution heavily simplifies the proof of K. Zhu in [20].
We are also interested here in some other possible equivalent characterizations of Schatten class Toeplitz operators. For this, we denote by ✷ z the natural extension to C n = R n + iR n of the wave operator ✷ x on the cone: We also have the following equivalent characterization. Again the condition p ≥ max{ n r −1 ν+ n r , 2 n r −1 ν+ n r +2m } is due to integrability conditions of the determinant function.
For the proof of Theorem 1.3, we derive the necessary condition for 1 ≤ p < ∞ and the sufficient condition for 0 < p < 1 from a more general result for any positive operator. The proof of the other parts essentially uses the properties of Bergman balls and the δ-lattices. We also refer to [15,16] for this type of results.
We are essentially motivated here by the idea of extending the results in [13] on Schatten class S p (A 2 ν (D)) for P ≥ 1, to the case 0 < p < 1, and settling the problem of the characterization of Schatten class S p (A 2 ν (D)) for 1 ≤ p < 2 for the Cesàro-type operator introduced in [13].
The paper is organized as follows. In the next section, we present some useful tools and results needed in the proofs of the above results . The proof of Theorem 1.2 is given in Section 3. In Section 4, we provide characterization of Schatten class for general positive operators. We prove Theorem 1.3 in Section 5. In the last section, we apply our results to extend to the range 1 ≤ p < 2, the characterization of Schatten class S p (A 2 ν (D)) for the Cesàro-type operator obtained in [13].
As usual, given two positive quantities A and B, the notation A B (resp. A B) means that there is an absolute positive constant C such that A ≤ CB (resp. A ≥ CB). When A B and B A, we write A ≍ B and say A and B are equivalent. Finally, all over the text, C, C k , C k,j will denote positive constants depending only on the displayed parameters but not necessarily the same at distinct occurrences. The same remark holds for lower case letters.

Preliminary results
In this section, we give some fundamental facts about symmetric cones, Berezin transform and related results.
2.1. Symmetric cones, Bergman metric and estimations of the determinant function. It is well known that a symmetric cone Ω induces in V ≡ R n a structure of Euclidean Jordan algebra, in which Ω = {x 2 : x ∈ V }. Let e be the identity element in V . Denote by G(Ω) the group of transformations of R n leaving invariant Ω. We recall that the group G(Ω) acts transitively on Ω. We denote by H the subgroup of G(Ω) that acts simply transitively on Ω, that is for x, y ∈ Ω there is a unique h ∈ H such that y = hx. Observe that if we still denote by R n the group of translation by vectors in R n , then the group G(D) = R n × H acts simply transitively on D.
Recall that δ > 0, is the Bergman ball centered at z with radius δ, where d(·, ·) is the Bergman distance (for a definition, see for example [13]). It well known that the measure We recall the following (see [2,Theorem 5.4]).
Moreover, there is an integer N (depending only on D) such that each point of D belongs to at most N of these balls.
We observe that We refer to [2, Theorem 5.6] for the following sampling theorem.
. Then the following assertions hold.
(1) There is a positive constant C δ such that every f ∈ A p ν (D) satisfies We have the following atomic decomposition with change of weight which is derived from [13,Theorem 3.2].
Assume that the operator P µ is bounded on L 2 ν (D) and let {ζ j } j∈N be a δ-lattice in D. Then the following assertions hold.
(i) For every complex sequence {λ j } j∈N in l 2 , the series is convergent in A 2 ν (D). Moreover, its sum f satisfies the inequality where C δ is a positive constant. (ii) For δ small enough, every function f ∈ A 2 ν (D) may be written as where C δ is a positive constant.
The following consequence of the mean value theorem (see [2]) is needed.
We will be using the following Korányi's lemma.
We close this subsection by recalling the following consequence of [2, Corollary 3.4] and the above Korányi's lemma.
Lemma 2.8. Let ν > n r − 1, δ > 0 and z, w ∈ D. There is a positive constant C δ such that for all z ∈ B δ (w), If δ is sufficiently small, then there is C > 0 such that for all z ∈ B δ (w), The following was first proved [7, Lemma 5.1] in the case of homogeneous Siegel domains of type II. Lemma 2.9. Let 0 < δ ≤ 1, α, β ∈ R with α > 2 n r − 1, β > 2 n r − 1 and α > β + n r − 1. Then for any ε > 0, there exists A ε > 0 such that if {z j = x j + iy j } is a sequence of points of D in a δ-lattice satisfying inf j =k d(z j , z k ) ≥ A ε , then for any integer j, the following estimate holds Proof. Let us give ourself A > 0. Thanks to Remark 2.2, we may assume that the sequence {z j = x j + iy j } is such that d(z j , z k ) ≥ A for all j = k. We first observe with Lemma 2.5 that .

It follows that
Let g ∈ G(D) be the transformation such that g(ie) = z j , and put w = g(ζ).
From the assumptions on α and β together with Lemma 2.6, one has that the integral r (ζ) converges. Hence, there exists A ε > 0 such that for all A ≥ A ε , the following inequality holds with C as in the above estimate of S. The proof is complete.

Averaging functions and Berezin transform.
The following was proved in [13] for 1 ≤ p ≤ ∞. A careful observation of the proof of [13,Lemma 2.9] shows that the result extends to 0 < p < 1.
belongs to L p ν (D).
Note that the above lemma allows flexibility on the choice of the radius of the ball. This fact is quite useful as seen in [13].
We have the following result.
2.3. Schatten class operators. In this subsection, H is a Hilbert space with associated norm · . The spaces of bounded and compact linear operators on H are denoted B(H) and K(H) respectively. We recall that any positive operator T ∈ K(H), then one can find an orthonormal set {e j } of H and a sequence {λ j } that decreases to 0 such that For 0 < p < ∞, we say a compact operator T with a decomposition as above belongs to the Schatten-Von Neumann p-class S p := S p (H), if For p = 1, we denote by S 1 = S 1 (H) the trace class. We recall that for T ∈ S 1 , the trace of T is defined by where {e j } is any orthonormal basis of the Hilbert space H.
It is known that a compact operator T on H belongs to the Schatten class S p if and only if the positive operator (T * T ) 1/2 belongs to S p , where T * denotes the adjoint of T . In this case, we have ||T || Sp = ||(T * T ) 1/2 || Sp . It is also well known that a positive T belongs to S p if and only if the operator T p belongs to the trace class S 1 . In this case, ||T || Sp = ||T p || S 1 .
We also recall that if T is a compact operator on H, and p ≥ 1, then that T ∈ S p is equivalent to j | T e j , e j | p < ∞ for any orthonormal set {e j } in H (see [18]).
The following can be found in [18]. We also observe the following (see [19]) Lemma 2.13. Let T be any bounded operator on H and assume that A is bounded surjective operator on H. Then T belongs to S p if and only if the operator A * T A belongs to S p .
Finally, we will need the following result (see [12]) Lemma 2.14. Let T be any bounded operator on H and let {e k } be an orthonormal basis of H. Then for any 0 < p ≤ 2, we have | T e k , e j | p .

Schatten class membership of Toeplitz operators
The aim of this section is to give criteria for Schatten class membership of Toeplitz operators on the weighted Bergman space A 2 ν (D).
3.1. Proof of Theorem 1.2. We start by proving the following.
Lemma 3.1. Let µ be a positive Borel measure on D, and ν > n r −1. Assume that n r −1 ν+ n r < p < 1. Suppose that for any δ-lattice (δ ∈ (0, 1)) {ζ j } j∈N in the Bergman metric of D, the sequence {μ δ (ζ j )} belongs to l p , that is Then the Toeplitz operator T µ belongs to the Schatten class S p (A 2 ν (D)). Moreover, Proof. Let σ be large enough so that P σ is bounded on L 2 ν (D). Thanks to Lemma 2.10, we can suppose that δ is small enough so that any f ∈ A 2 ν (D) can represented as in Theorem 2.4. That is Let {e k } k≥1 be a fixed orthonormal basis on A 2 ν (D). Consider the operator S : . Then it follows from Theorem 2.4 that S is a bounded and surjective operator on A 2 ν (D). We know from Lemma 2.13 that T µ belongs to S p (A 2 ν (D)) if and only if T = S * T µ S belongs to S p (A 2 ν (D)). It follows from Lemma 2.12 that we only have to prove that We first observe that Hence using Lemma 2.8, we obtain Recalling that 0 < p < 1, we then obtain Using the fact that each point in D belongs to at most N balls B k and the condition n r −1 ν+ n r < p < 1, we obtain using Lemma 2.6, the following for the inner sum Using the latter, we conclude that We next prove the reverse of the above result.
Lemma 3.2. Let µ be a positive measure on D. Assume that T µ ∈ S p (A 2 ν (D)) for some 0 < p < 1. Let {ζ j } j∈N be a δ-lattice in D. Then the sequence {μ δ (ζ j )} belongs to l p . Moreover, Proof. We start by considering σ large enough so that σ + n r and σ + n r − 1 2 (ν + n r ) satisfy the conditions in Lemma 2.9. Let ε > 0, and let A ε be as in Lemma 2.9. Following Remark 2.2, we may assume that our sequence {ζ j } is such that d(ζ j , ζ k ) > A ε for j = k. We further assume that A ε is large enough so that corresponding balls B k are disjoint. Consider the following measure: Then 0 ≤ ω ≤ µ, ω = µ on each ball B k . We also have the inequality T ω p Sp ≤ T µ p Sp . Now as in the proof of the previous result, we fix an orthonormal basis {e k } of A 2 ν (D) and consider the same operator S defined on A 2 ν (D) by S(e k ) = f k with f k (z) = K σ (z, ζ k )∆ σ+ n r − 1 2 (ν+ n r ) (ℑζ j ). We recall with Theorem 2.4 that S is bounded and surjective on A 2 ν (D). Put again T = S * T ω S. Then as T ω ∈ S p (A 2 ν (D)), T also belongs to S p (A 2 ν (D)) and we have and R = T − D. We observe that Hence, if we can prove that with c 2 as small as we want, then the proof will be completed. We start by estimating the diagonal operator D. As D is positive, we have That is We now turn to the estimation of R p Sp . First, using Lemma 2.12, we obtain As the balls B l are disjoint, using Lemma 2.8, we obtain ≍ l |f j (ζ l )||f k (ζ l )|∆ ν+ n r (ℑζ l )μ δ (ζ l ).
V . We may suppose that µ(B 1 (ie)) > 0 (if not change the radius of the Bergman ball). Then using Lemma 2.7, we obtaiñ It follows that ifμ(z) ∈ L p (D, dλ), then we should have which by Lemma 2.6 is possible only if p(ν + n r ) > 2 n r − 1. Now the equivalences (ii)⇔(iii)⇔(iv) are from Lemma 2.11. The equivalence (i)⇔(ii) is derived from Lemma 3.1 and Lemma 3.2. The proof is complete.
3.2. Schatten class for general operators. We consider here Schatten class criteria for an arbitrary operator defined on A 2 ν (D) with values in a Hilbert space H. We denote by B(A 2 (D), H) the set of bounded operators from A 2 ν (D) to H. To avoid any confusion, we denote by ·, · H and ·, · ν the inner products in H and A 2 ν (D) respectively. We start with the Hilbert-Schmidt class S 2 := S 2 (A 2 ν (D), H). Proof. This result was proved in [17]. As the definition of Bergman spaces here is quite different, let us give a proof here for completeness. Let {e j } is an orthonormal basis of H, then ||T * e j || 2 A 2 ν = C n,m ||T * || 2 S 2 = C n,m ||T || 2 S 2 .
In the fourth equality, we used the fact that ✷ m z is an isometric (up to constant C n,m ) isomorphism from A 2 ν (D) onto A 2 2m+ν (D).
We will deduce some results from the above one. The first one is the following which follows as in [17, Lemma 3.2].
3.3. Proof of Theorem 1.3. We start by observing that taking T = T µ in Proposition 3.5, we obtain the following reproducing kernel thesis for T µ . 1 ≤ p < 2. We refer to [8,12,19,20] for the corresponding results on some classical domains.
Let us recall that the Besov space B p (D) is the subset of H n (D) consisting of functions f such that ∆ n ✷ n f ∈ L p (D, dλ) = L p (D, dV (z) ∆ 2 n r (ℑz) ). For more on Besov spaces of tube domains over symmetric cones, we refer the reader to [3,4].
We now obtain the following.
Theorem 4.1. Let 1 ≤ p < 2, ν > n r − 1. If g is a given holomorphic function in D, then the Cesàro-type operator T g belongs to S p (A 2 ν (D)) if and only if g ∈ B p (D).