Schatten class operators on exponential weighted Bergman spaces

In this paper, we study Toeplitz and Hankel operators on exponential weighted Bergman spaces. For 0<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p<\infty $\end{document}, we obtain sufficient and necessary conditions for Toeplitz and Hankel operators to belong to Schatten-p class by the averaging functions of symbols. For a continuous increasing convex function h, the Schatten-h class Toeplitz and Hankel operators are also characterized.


Introduction
Let D = {z ∈ C | |z| < 1} be the unit disk in the complex plane C and dA(z) = dxdy π be the normalized Lebesgue area measure on D. Let L denote a class (see [2,13] for more details about the class).A function ρ(z) is said to be in L if ρ(z) is positive on D satisfying the following conditions: (a) For any z ∈ D, there is a constant c 1 > 0 such that ρ(z) ≤ c 1 (1 -|z|).
Write A B for two quantities A and B if there is a constant C > 0 such that A ≤ CB.Furthermore, A B means that both A B and B A are satisfied.A subharmonic function ϕ(z) ∈ C 2 (D) satisfying ( ϕ(z)) -1/2 ρ(z) is called ϕ ∈ L * , where ρ(z) ∈ L and is the standard Laplace operator.
The Lebesgue space L p ϕ (0 < p < ∞) consists of all measurable functions f on D such that In particular, L ∞ ϕ consists of all measurable functions f on D such that f ϕ,∞ = esssup z∈D f (z)e -ϕ(z) < ∞. © The Author(s) 2023.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Notice that A 2
ϕ is a reproducing kernel Hilbert space, and hence there is a function K ϕ,z ∈ A 2  ϕ such that the orthogonal projection P from L 2 ϕ to A 2 ϕ can be represented as See [3,13].The function K ϕ,z (•) is called the reproducing kernel of Bergman space A 2 ϕ and has the property that K ϕ,z (w) = K ϕ,w (z) for every z, w ∈ D. It follows from [3, Theorems 4.1 and 4.2] that, for ϕ ∈ E and 1 ≤ p ≤ ∞, the Bergman projection P : For a positive Borel measure μ on D and a measurable function f , the Toeplitz operator and Hankel operator are defined respectively by T μ (g)(z) = D g(w)K ϕ (z, w)e -2ϕ(w) dμ(w), g ∈ A p ϕ and H f (g)(z) = D f (z)g(w)f (w)g(w) K ϕ (z, w)e -2ϕ(w) dA(w), g ∈ A p ϕ .
The pioneering work on this class of exponential weighted Bergman spaces was done by Oleinik and Perelman [14].Throughout this paper, we call these spaces OPS.Later, has attracted much attention.In [12], Lin and Rochberg characterized the boundedness and compactness of Hankel operators on exponential weighted Bergman spaces.To further study these spaces, Lin and Rochberg [13] gave the necessary and sufficient conditions for Schatten-p class Toeplitz (or Hankel) operators when 1 ≤ p < ∞.Furthermore, for 0 < p < 1, the sufficient condition for Schatten class membership of the Toeplitz operator was obtained as well.In [3,4], Arroussi and Pau studied the dual space and estimates of the reproducing kernel.
Borichev, Dhuez, and Kellay [5] introduced another exponential weighted Bergman spaces.The authors, in [2], showed the Schatten class membership of the Toeplitz operator on spaces introduced by [5].Hu, Lv, and Schuster [8] characterized a new kind of space, which contains these exponential weighted Bergman spaces considered in [5], write HLS for simplicity.Indeed, the spaces HLS differ from the spaces in this paper, see [8].In [9], Hu and Pau gave bounded and compact Hankel operators associated with general symbols.Zhang, Wang and Hu [17] showed the boundedness and compactness of Toeplitz operators with positive symbols acting between different spaces HLS, and Schatten-p class membership.Recently, in [16], the authors studied the sufficient and necessary conditions for Schatten-p class membership of Hankel operators associated with general symbols on HLS.
For 0 < p < ∞, by using averaging functions, we obtain the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive symbols and Hankel operators with general symbols on OPS.These results fill the research gap of [13].Generally speaking, the difficulty in such problems lies in the characterization of 0 < p < 1.For this goal, we need more tools than [13].Schatten-h class membership of operators is an important generalization of Schatten-p class operators, and it is interesting to study Schatten-h class membership.We refer to [1] and the relevant references therein for a brief account on Schatten-h class.In this paper, we explore Schatten-h class Toeplitz and Hankel operators on the spaces.Such properties of Hankel operators are not yet known in the existing literature.
By [8,Theorem 3.2], the following estimate holds for the reproducing kernel in this space: there exist constants C, σ > 0 such that where d ρ (z, w) is the Bergman metric induced by reproducing kernel.However, the reproducing kernel in OPS does not have the similar estimate, which brings more obstacles to the research in this paper.The paper is organized as follows.In Sect.2, we give some basic notation and lemmas.In Sect.3, we show the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive symbols, and give the characterization for Schatten-h class membership of Toeplitz operators induced by continuous increasing convex functions.Finally, in Sect.4, we investigate membership in Schatten-p class Hankel operators with general symbols, and also obtain Schatten-h class properties of Hankel operators.

Preliminaries
We begin with giving some basic notation and lemmas.For z ∈ D and r > 0, let D(z, r) = {w : |w -z| < r} be the Euclidean disk with radius r and center z.Also, we use D r (z) = D(z, rρ(z)) to denote the disk with radius rρ(z) and center z.
It is from [3, Lemma A] that we have the following pointwise estimate.
Then there exists a constant M ≥ 1, for f ∈ H(D) and small enough δ > 0, such that As we known, the covering lemma is useful for studying Bergman spaces, so does exponential weighted Bergman spaces.The following lemma comes from [2, Lemma B].
Lemma 2.3 Suppose ρ ∈ L and 0 < r < m ρ .Then there exists a sequence j=1 is a covering of D of finite multiplicity, that is, for any z ∈ D, where N is a positive constant integer.
A sequence {a j } ∞ j=1 satisfying the above lemma is called the (ρ, r)-lattice.Furthermore, the conditions (a) and (c) indicate there is a s > 0 such that It is important to investigate pointwise and norm estimates of the reproducing kernels K ϕ,z on A 2 ϕ .The following results are from [3, Lemma B, Theorem 3.1 and (3.1)].If ϕ ∈ L * , 0 < r < m ρ and w ∈ D r (z), then we have With the help of estimates for the reproducing kernels, we get the following atomic decomposition.
To describe the Schatten-p membership of Hankel operators, we need some auxiliary conclusions.For z, w ∈ D, we write .
Given a positive Borel measure μ on D and r > 0, the averaging function μr with respect to measure μ is defined by where g ∈ H(D).

Schatten class Toeplitz operators
In this section, for 0 < p < ∞, we investigate the sufficient and necessary conditions for Schatten-p class membership of Toeplitz operators with positive measure symbols on OPS.Also, we give the characterization for Schatten-h class membership of Toeplitz operators where h is a continuous increasing convex function.
Let T : H 1 → H 2 be a bounded linear operator, and write s j (T) for the singular values of T, where Here rank(K) means the rank of operator K .Recall that the operator T is compact if and only if s j (T) → 0 whenever j → ∞.
and we write T ∈ S p (H 1 , H 2 ).Futhermore, • S p is actually a norm when 1 ≤ p < ∞ and Using and it is easy to see T ∈ S p if and only if T * T ∈ S p 2 .As we known, the Schatten class of Toeplitz operators with positive measure symbols is an important problem in operator theory, which has been described in many papers (see, for example, [2,13,17]).The following theorem is closely related to the main result [2, Theorem 1.2].To Study the Schatten class of Toeplitz operators, we define the measure dλ ρ by and μ is a finite positive Borel measure on D.
Then following statements are equivalent: Proof The proof of (b) ⇔ (c) ⇔ (d) is similar to [17, Proposition 2.5], and we omit the details here.Indeed, this proof indicates the L p behavior of averaging function μr is independent of r. (That is, for small enough r, μδ L p μr L p with small enough δ.)The rest part is an analogue of [17, Theorem 5.1], and for the convenience of readers, we give the proof for implication (a) ⇒ (c) when 0 < p < 1.
Assume the Toeplitz operator T μ is in S p (A 2 ϕ ).Let {w n } be a (ρ, r)-lattice with r ∈ (0, m ρ ] sufficiently small.Set a large enough integer m ≥ 2, by Lemma 2.9, the lattice {w n } can be devided into subsequences such that where w i and w j are in the same subsequence.Let {a n } be such a subsequence, and measure ν be defined by where χ n is the characteristic function of D r (a n ).Disks D r (a n ) are pairwise disjoints since m ≥ 2. Note that T μ ∈ S p (A 2 ϕ ) and 0 ≤ ν ≤ μ, thus 0 ≤ T ν ≤ T μ , and then T ν ∈ S p (A 2 ϕ ) and we have We now take a decomposition of the operator T as T = T 1 + T 2 , where T 1 is the diagonal operator defined by and T 2 = T -T 1 is the non-diagonal part.Using Rotfel'd inequality (see [15]), we see Notice that T 1 is a positive diagonal operator, this together with the definition of ν, (2.1), (2.4), and (2.5) gives For 0 < p < 1, [18, Proposition 1.29] and Lemma 2.3 show . Hence, for ξ ∈ D r (a j ), we get either Therefore, for any ξ ∈ D r (a j ), we may assume For any n, k ∈ N + , set This, combined with (3.7), yields Let M be large enough.Here M is from Theorem 2.6.Applying And hence, It follows from (2.4), (2.5), and (2.6) that By joining (3.9), (3.10), and Lemma 2.1, we obtain Applying Lemmas 2.1, 2.2, and 2.3 (c), for ξ ∈ D r (a j ), we conclude p/2 e -pϕ(z)/2 dA(z) where k ϕ,a n (z) p/2 e -pϕ(z)/2 dA(z).

The analogous reasons indicate
So, for M large enough, we have And hence, for 0 < p < 1, ) can be estimated further as On the other hand, by the definition of S k (a j ), we see p/2 e -pϕ(z)/2 dA(z). (3.12) We claim that For this goal, by (2.4), (2.1), and Lemma 2.3 (d), for some r 0 > 0, we get Taking M in Theorem 2.6 such that Mp/2 -2 > 0, then By Lemma 2.3, for j = 0, 1, 2, . . ., when a k ∈ D r 0 2 j+1 (z), we obtain So and hence (3.13) holds by (3.14) and the following estimate Bearing in mind (3.13), (3.12) can be estimated as This together with (3.6) and (3.5) yields Since the above estimate holds for each of the subsequences {w n }, we finally obtain ), which finishes this proof.
We are going to describe the Schatten-h class Toeplitz operators.See [1] and the references therein for details about the Schatten-h class.We give first the following analogous definition.

Definition 3.2
Let T be a compact operator and h : R + → R + be a continuous increasing convex function.We say that T ∈ S h if there is a positive constant c such that Similar to [1], we get the following consequence.Theorem 3.3 Suppose h : R + → R + is a continuous increasing convex function, and μ is a positive Borel measure such that Toeplitz operator T μ : A 2 ϕ → A 2 ϕ is compact.Then T μ ∈ S h if and only if there exists a constant c > 0 such that k=1 be an orthonormal basis for A 2 ϕ , and where s k is the singular value sequence of T μ .With the help of the convexity of h, Jensen's inequality, (2.4), and (2.5), we have Conversely, if there exists c > 0 such that D h(c μ(z))ρ -2 (z) dA(z) < ∞, then it follows from (2.4) and (2.5) that 2 e -2ϕ(w) dμ(w) ≤ μ(z).

Notice that
then by Jensen's inequality again we get which gives T μ ∈ S h .This completes the proof.

Schatten class Hankel operators
This section devotes to studying membership in Schatten ideals of Hankel operators with general symbols.First, when 0 < p < ∞, we get the sufficient and necessary conditions for Hankel operators are in Schatten-p class.Here we mainly discuss case 0 < p < 1, see case 1 ≤ p < ∞ in [13].Next, for a continuous increasing convex function h, we obtain the sufficient and necessary conditions for Hankel operators to be in Schatten-h class.This kind of problem is new for Hankel operators.Proof See [6].
Let L 2 loc (D) denote the space consists of locally square integrable Lebesgue measurable functions on D.
where H(D r (z)) is the analytic functions space on D r (z).For z ∈ D, f ∈ L 2 (D r (z), dA) and r > 0, the averaging function of |f | on D r (z) is defined by Lemma 4.2 For z ∈ D, f ∈ L 2 (D r (z), dA), and r > 0, there exists an h ∈ H(D r (z)) such that Proof The proof is similar to [11,Lemma 3.3].
For z ∈ D and r > 0, let denote the Bergman space on D r (z).Let B z,r denote Bergman projection induced by the reproducing kernel of A 2 (D r (z), dA).As we known, B z,r is bounded and B z,r h = h, where h ∈ A 2 (D r (z), dA).The following consequence is similar to [11,Lemma 3.4] with q = 2.

Lemma 4.3
For z ∈ D and r > 0, if f ∈ L 2 (D r (z), dA), then we have

.3)
Proof Taking h from Lemma 4.2, we have h ∈ A 2 (D r (z), dA) since f ∈ L 2 loc (D).Then B z,r h = h.By trigonometric inequality and Lemma 4.2, , and hence this proof is complete.
Definition 4.4 By the decomposition above, we define Note that f 1 (z) is actually a finite summation for any z ∈ D, and by supp ψ j ⊆ D r/3 (a j ) ⊆ D r (a j ), then f 1 is well-defined.
where z ∈ D and C > 0 is independent of f .
Proof Since h j ∈ H(D r (a j )) and ψ j ∈ C ∞ (D), f 1 ∈ C 1 (D).For z ∈ D, without loss of generality, we may assume z ∈ D r/3 (a 1 ).It is easy to check that D r/9 (z) ⊆ D r (a j ) whenever z ∈ D r/3 (a j ).By ∞ j=1 ∂ψ j (z) = 0 and the subharmonic property of If z ∈ D r/3 (a j ), then we have D r (a j ) ⊆ D 9r (z), and Hence, If w ∈ D r/9 (z), then D 3r (w) ⊆ D 9r (z).Thus, similar to (4.6), Using Cauchy-Schwarz inequality, Therefore, This finishes the proof.
Proof The proof is an analogue of [17,Proposition 2.4].
By [13], Hence, the condition G r (f ) ∈ L ∞ is natural in the study of Schatten class membership of Hankel operators.
Then both H f 1 and H f 2 are bounded, and Proof See [13, Theorem 3.1].Now we are ready for the characterization of Schatten class Hankel operators.
Theorem 4.9 Suppose ϕ ∈ E, 0 < p < ∞, 0 < r ≤ m ρ and H f is densely defined satisfying G r (f ) ∈ L ∞ .Then following statements are equivalent: (e) For some (or any) (ρ, r)-lattice Proof (a) ⇒ (b).We give only the case 0 < p < 1.Let {a j } ∞ j=1 be a (ρ, r)-lattice.By Lemma 2.9, {a j } ∞ j=1 can be devided into N subsequences, if a i and a j are in the same subsequence, then In fact, just consider one of subsequences here.Without loss of generality, it is assumed that {a j } ∞ j=1 .For any finite subset J ⊆ N + , let {e j } ∞ j=1 be an orthonormal basis for A 2 ϕ , and Then, by Parseval's equality, g, e j 2 = g 2 ϕ .
It follows from Lemma 2.7 that A is bounded on A 2 ϕ .If χ D r (a j ) gk ϕ,a j -P a j ,r (gk ϕ,a j ) L 2 ϕ = 0, we let and Let {c j } j∈J denote nonnegative sequence, we define the operator B by It is easy to check that B is bounded on A 2 ϕ , and B ≤ sup j∈J {c j }.It follows that The application of Lemma 4.1 gives Taking a decomposition of the operator BH f A as the diagonal part and the non-diagonal part Let i, j ∈ J and i = j.Then there exists w j,i ∈ D r (a j ) such that This combined with (2.6) and (2.1), for z ∈ D r (a j ), implies We claim that |w j,ia i | ≥ 2 k-2 r min(ρ(w j,i ), ρ(a i )).If not, we assume |w j,ia i | ≤ 2 k-2 r × min(ρ(w j,i ), ρ(a i )).By (2.1) and the trigonometric inequality, So |a ja i | < 2 k r min(ρ(a j ), ρ(a i )), which causes a contradiction with (4.11).Thus, for z ∈ D r (a j ), (4.13) By joining (4.12) and (4.13), we obtain Set r 0 = 3r.Fix j ∈ J, then, for any z ∈ D r (a j ), we have It is from (2.1) that, for any z ∈ D r (a j ),
Hence, for any J, (d) ⇔ (e).See Lemma 4.6.(d) ⇒ (a).To finish this, we let M f 2 and M ρ ∂f 1 denote multiplication operators.Let φ be f 2 or ρ ∂f 1 .By G r (f )(z) ∈ L ∞ and Lemma 4.6, M r (φ)(z) ∈ L ∞ .We next show the operator M φ is bounded from A 2 ϕ to L 2 ϕ .Indeed, by Lemma 2.10 with p = 2, then for g ∈ A 2 ϕ we have   Here {ψ j } ∞ j=1 is the unit decomposition induced by {D Now let H(D) be the space of analytic functions in the unit disk D. The exponential weighted Bergman spaces A p ϕ = L p ϕ ∩ H(D).When 1 ≤ p ≤ ∞, A p ϕ is a Banach space, and A p ϕ is a Fréchet space if 0 < p < 1.

Lemma 2 . 8 ([ 2 ,
Lemma 4.4]) Let ρ ∈ L and {a j } j be a (ρ, r)-lattice on D. Then for any w ∈ D, the setD m (w) = z ∈ D | d ρ (z, w) < 2 m rcontains at most K points of the lattice, where K depends on the positive integer m, but not on the point w.Lemma 2.9 ([2, Lemma 4.5]) Let ρ ∈ L, r ∈ (0, m ρ ] and k ∈ N + .Any (ρ, r)-lattice {a j } ∞ j=1 on D, can be partitioned into M subsequences such that, if a i and a j are different points in the same subsequence, then

Lemma 4 . 6
Let 0 < p < ∞ and f ∈ L 2 loc (D).Then following statements are equivalent: (a) For some (or any r (f )(a j ) p sup j∈J c p j .Therefore (a) ⇒ (b) since l ∞ is the dual space of l 1 .

m |z -w|. Theorem 2.6 If ϕ ∈ E, then for any M ≥ 1 there is a constant C > 0 such that
h j g, e i e j , D r (a j ) fk ϕ,a j -P a j ,r (fk ϕ,a j ) p S p .By Lemma 2.4, there exists a constant C > 0 such that for z ∈ D r (a j ) if and only if M r (φ)(z) ∈ L p (D, ρ -2 dA), and soM φ ∈ S p .Since H f 1 (g) ϕ,2 gρ ∂f 1 ϕ,2 and H f 2 (g) ϕ,2 f 2 g ϕ,2, both H f 1 and H f 2 are in S p , therefore H f ∈ S p .This finishes the proof.