Hankel Operators on Standard Bergman Spaces

We study Hankel operators on the standard Bergman spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A^2_{\alpha},\,\alpha > -1}$$\end{document}. A description of the boundedness and compactness of the (big) Hankel operator Hf with general symbols \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \in L^2(\mathbb{D}, dA_{\alpha})}$$\end{document} is obtained. Also, we provide a new proof of a result of Arazy–Fisher–Peetre on the membership in Schatten p-classes of Hankel operators with conjugate analytic symbols.


Introduction
If T is an operator induced (in some way) by a symbol f going from some Hilbert space to another Hilbert space, one is going to hope that individual properties of the symbol (such as smoothness or growth conditions) will give information on the properties of the operator (boundedness, compactness, or membership in Schatten-Von Neumann ideals). In the present paper, we will study this when dealing with Hankel operators on standard weighted Bergman spaces. For α > −1, the weighted Bergman space A 2 α consists of those functions f analytic on the unit disk D such that where d A α (z) = (α + 1)(1 − |z| 2 ) α d A(z) and d A is the normalized area measure on D. The space A 2 α is a Hilbert space with reproducing kernel given by K z (w) = (1 −zw) −2−α ; it is also a closed subspace of L 2 (D, d A α ), and the orthogonal projection from L 2 (D, d A α ) to A 2 α is given by Given a function f ∈ L 2 (D, d A α ), the Hankel operator with symbol f is the linear operator H f : A 2 α → L 2 (D, d A α ) defined by The operator H f is densely defined on A 2 α . For example, it is well defined in H ∞ , the algebra of all bounded analytic functions on D. The following integral formula is very useful when one is going to estimate the norm of a Hankel operator: It has been a lot of activity in the theory of Hankel operators on Bergman spaces in recent years, and this topic has become a classical theme in complex analysis and operator theory (see for example [1,3,4,10,11,13], and [17]). For Hankel operators with conjugate analytic symbols, that is Hf with f ∈ A 2 α , one has that Hf is bounded on A 2 α if and only if the symbol f belongs to the Bloch space; Hf is compact if and only if f belongs to the little Bloch space (see [1,2]); and the membership in Schatten p-classes of the Hankel operator Hf is equivalent to the function f being in the analytic Besov space B p for 1 < p < ∞, and to f being constant when 0 < p ≤ 1. Therefore, for conjugate analytic symbols, the picture on the boundedness, compactness and Schatten p-classes is complete. However, the proof of the necessity and sufficiency of the condition f ∈ B p for Hf being in the Schatten class S p when 1 < p < 2 given in [1] is rather difficult and technical, and it is our aim to provide a more "elementary" proof of that result. Recall that, for 1 < p < ∞, the analytic Besov space B p consists of those functions f analytic on D for which Also, if H and K are separable Hilbert spaces, a compact operator T from H to K is said to belong to the Schatten class S p if its sequence of singular numbers is in the sequence space p . Recall that the singular numbers of a compact operator T are the square root of the eigenvalues of the positive operator T * T , where T * denotes the adjoint of T . Also, the compact operator T admits a decomposition of the form where {λ n } are the singular numbers of T, {e n } is an orthonormal set in H , and { f n } is an orthonormal set in K. For p ≥ 1, the class S p is a Banach space with the norm T p = n |λ n | p 1/ p , while for 0 < p < 1 one has the inequality S + T p p ≤ S p p + T p p . We refer to [17,Chapter 1] for a brief account on the theory of Schatten p-classes.
We consider also the study of the boundedness, compactness and membership of Schatten p-classes of the Hankel operator H f for general symbols f ∈ L 2 (D, d A α ). In order to state the next result we need to introduce some notation. For z ∈ D and r > 0, let denote the hyperbolic disk with center z and radius r . Here β(z, w) is the Bergman or hyperbolic metric on D. Also, for any Lebesgue measurable set E in D, we use the notation |E| α := E d A α for the d A α -measure of E. Theorem 2 Let α > −1 and f ∈ L 2 (D, d A α ). The following conditions are equivalent: (c) For any (or some) r > 0, the function F r defined by , and f 2 has the property that for any (or some) r > 0 the function G r defined by is bounded on D.
The case α = 0 of Theorem 2 was proved by Luecking in [10], who also noticed that the same proof also applies to the case −1 < α < 1, and that the only missing part for the weighted case is a proof of the implication (d) implies (a), and this will be our contribution. The proof uses ∂-techniques, and the main ideas for the proof were essentially given in [7] (see also the related papers [6] and [8] of the same authors), where the corresponding result for a class of weighted Bergman spaces is obtained. However, the standard weights (1 − |z| 2 ) α are in the class considered in [7] only for α > 2, and also the statement that they give is for symbols in L 2 (D). We remark that Zhu in p. 233 of his book [17] considers the question of describing the boundedness of Hankel operators with general symbols on weighted Bergman spaces as an open problem. Here we will use the appropriate modifications in order to obtain a unified proof for all α > −1. We also obtain the corresponding analogues for compactness and membership in Schatten-Von Neumann ideals.
Throughout the paper, the letter C will denote an absolute constant whose value may change at different occurrences. The paper is organized as follows: Section 2 is devoted to some preliminaries needed for the proofs of the main results. A proof of Theorem 1 is given in Sect. 3, and we study the boundedness, compactness and membership in Schatten classes of Hankel operators with general symbols f ∈ L 2 (D, d A α ) in Sect. 4. Finally, we look at little Hankel operators in Sect. 5.

Preliminaries
We will use the fact that for any orthonormal set {e n } of A 2 α , one has n |e n (z) with equality if {e n } is also an orthonormal basis. The following integral estimate (see [17]) has become indispensable in this area of analysis, and will be used repeatedly throughout the paper.
We also need some well known variants of the previous lemma. First recall that a sequence {z n } of points in the unit disk D is said to be separated in the Bergman metric if there is a constant δ > 0 such that β(z j , z k ) ≥ δ for all j and k with j = k. In particular, there is a constant r > 0 such that the hyperbolic disks D(z k , r ) are pairwise disjoint.

Lemma 2.2
Let {z k } be a separated sequence in D, and let 1 < t < s. Then

Lemma 2.3 Let c > 0 and t > −1. Then
The following solution of the ∂-equation will be a key ingredient in the proof of Theorem 2.
provided the right hand-side integral is finite.
Proof The function u clearly satisfies the equation ∂u = f in D. The corresponding estimate will follow from Hölder's inequality and Lemma 2.
Thus, Fubini's theorem and Lemma 2.3 gives We also need the concept of an r -lattice in the Bergman metric. Let r > 0. A sequence {a k } of points in D is called an r -lattice if the unit disk is covered by the Bergman metric disks {D(a k , r )}, and β(a i , a j ) ≥ r/2 for all i and j with i = j. If {a k } is an r -lattice in D, then it also has the following property: for any R > 0 there exists a positive integer N (depending on r and R) such that every point in D belongs to at most N sets in {D(a k , R)}. There are elementary constructions of r -lattices in D.
A positive Borel measure μ in the unit disk is a Carleson measure for A 2 α if there exists a finite positive constant C such that It is well known (see [9], or Theorems 7.4 and 7.7 in [17] for example) that the Carleson measures for A 2 α are characterized by the condition describes the vanishing Carleson measures for A 2 α .

Proof of Theorem 1
Let f ∈ A 2 α and 1 < p < 2. We prove the necessity first. So, suppose that the Hankel operator Hf belongs to S p . We must show that f ∈ B p . Let β = (2 + α) p. By Theorem 5.21 of [17], f ∈ B p if and only if If K z is the reproducing kernel of A 2 α , since f K w ∈ A 2 α for each w ∈ D, an easy computation gives On the other hand, let Hf = n λ n ·, e n α f n be a decomposition of the operator Hf , with {λ n } being the singular numbers of Hf , and {e n }, { f n } are orthonormal sets in A 2 α and L 2 (D, d A α ) respectively. Then Hence, by Hölder's inequality, This, together with the fact that K z Since Hf p S p = n |λ n | p , it is enough to show that for some positive constant C independent of n. Now, since p < 2, we can use Hölder's inequality with exponent 2/ p > 1 to obtain where γ = p(4+α) 2− p . Now we use Hölder's inequality once again and Lemma 2.1 to obtain Therefore, since f n L 2 (D,d A α ) = 1, an application of Fubini's theorem and Lemma 2.1 yields This finishes the proof of the necessity part. Now we proceed to prove the sufficiency of the condition f ∈ B p . Since B p is included in the little Bloch space, it follows that Hf is compact. We want to show that the Hankel operator Hf is in the Schatten class S p or, equivalently, that the positive operator S = H * f Hf belongs to S p/2 . To this end, we fix a sufficiently large number b and use the atomic decomposition of A 2 α (see [17,Theorem 4.33]) to find a separated sequence {z n } in D such that A 2 α consists exactly of functions of the form and g α ≤ C λ 2 for some positive constant C independent of {λ n }.   Let {a n } ⊂ D be an r -lattice in the Bergman metric. Since |h n (z)| is comparable to |h n (a k )| for z ∈ D(a k , r ) we obtain Also, the subharmonicity of | f | p and the fact that D(z, r ) ⊂ D(a k , 2r ) for z ∈ D(a k , r ) gives [17,Proposition 4.13]). Recall that dλ(w) = (1 − |w| 2 ) −2 d A(w) is the hyperbolic metric in D. This together with the fact that p/2 ≤ 1 yields Finally, since {z n } is a separated sequence, it follows from Lemma 2.2 that This, together with the fact that |D(a k , r )| α (1 − |a k | 2 ) 2+α gives This completes the proof of the theorem.

Hankel Operators with General Symbols
Proof of Theorem 2 The implications (a) ⇒ (b); (b) ⇒ (c); and (c) ⇒ (d) follows the same arguments of the proof given in [10] (see also the proof of Theorem 8.34 in K. Zhu's monograph [17]). So, suppose that (d) holds. The condition on f 2 says that | f 2 | 2 d A α is a Carleson measure for A 2 α . Therefore and the Hankel operator H f 2 is bounded on A 2 α . It remains to show that the Hankel Since completing the proof of the boundedness of H f .
The corresponding result for compactness is the following one.

Theorem 3
Let α > −1 and f ∈ L 2 (D, d A α ). The following conditions are equivalent: (c) For any (or some) r > 0, the function F r defined by , and f 2 has the property that for any (or some) r > 0 the function G r defined by Proof Since every function in C 0 (D) is bounded in D, each of the conditions implies the boundedness of the Hankel operator H f on A 2 α . We will prove only the implication (d) ⇒ (a), since the other implications are well known. Thus, suppose that f = f 1 + f 2 with (1−|z| 2 ) ∂ f 1 (z) ∈ C 0 (D), and | f 2 | 2 d A α being a vanishing Carleson measure for A 2 α . We will show that both H f 1 and H f 2 are compact. The condition on f 2 implies that the multiplication operator M f 2 is compact from A 2 α into L 2 (D, d A α ) (see Theorem 7.8 in [17]), and therefore the Hankel operator H f 2 = (I − P α )M f 2 is compact. To show that H f 1 is compact, let {g n } be a bounded sequence in A 2 α that converges to 0 uniformly on compact subsets of D. Given any ε > 0, there exists 0 < r < 1 such that and we can choose a positive integer n 0 with |g n (z)| < ε, |z| ≤ r, n ≥ n 0 .
From the proof of Theorem 2 it follows that ≤ Cε for all n ≥ n 0 . This proves that lim n H f 1 g n α = 0, so H f 1 is compact from A 2 α into L 2 (D, d A α ) finishing the proof of the theorem.
The following result characterizes the membership of the Hankel operator H f in the Schatten classes S p for p ≥ 1. Recall that dλ(z) = (1 − |z| 2 ) −2 is the hyperbolic measure on D.

Theorem 4 Let p ≥ 1 and f ∈ L 2 (D, d A α ) such that H f is bounded on A 2
α . The following conditions are equivalent: belongs to L p (D, dλ). (c) For any (or some) r > 0, the function F r defined in Theorem 2 belongs to L p (D, dλ).
(d) f admits a decomposition f = f 1 + f 2 , where for any (or some) r > 0 the function belongs to L p (D, dλ), and the function G r defined in Theorem 2 also belongs to L p (D, dλ).
Remark For p ≥ 2, condition (b) can be replaced by the condition (b2) The function z → H f k z α belongs to L p (D, dλ), and (d) can also be replaced by the condition belongs to L p (D, dλ) and for any (or some) r > 0 the function G r defined in Theorem 2 also belongs to L p (D, dλ).
Proof Apart from the implication (d) implies (a), all the other implications are known (see [10] or [17,Theorem 8.36]). So, suppose that (d) holds, that is, where the functions H r and G r both belong to L p (D, dλ). Since for any g ∈ H ∞ one has , it suffices to show that the multiplication operator M ψ : A 2 α → L 2 (D, d A α ) belongs to S p for ψ = f 2 or ψ = h. This is equivalent to M * ψ M ψ being in S p/2 , and since M * ψ M ψ = T |ψ| 2 where T ϕ denotes the Toeplitz operator with symbol ϕ, by Theorem 7.18 of [17] the conditions in (d) are exactly what is needed to have both T | f 2 | 2 and T |h| 2 belong to S p/2 . Thus the corresponding multiplication operators M f 2 and M h are in S p finishing the proof of the theorem.

Little Hankel Operators
Let A 2 α be the space of conjugate analytic functions in The operator h f is unbounded in general. However, h f is bounded if f is bounded, and we clearly have h f ≤ f ∞ . In the study of little Hankel operators, it turns out that it is more convenient to study hf instead of h f . Throughout this section, let V α be the integral operator defined by Recall that k z are the normalized reproducing kernels of A 2 α . For a given function f ∈ L 2 (D, d A α ), one has the identity hf = h c α V α f for some positive constant c α depending only on α, in the sense that hf g = h c α V α f g for all g ∈ H ∞ , which is dense in A 2 α (see [17,Chapter 8]). The properties of V α f can be used in order to obtain descriptions of the boundedness, compactness and membership in Schatten classes of the little Hankel operator hf . In fact, it is proved in [5] and [14] that hf is bounded on , and the compactness is characterized by the condition V α f ∈ C 0 (D). The corresponding description for the membership in the Schatten classes S p with p ≥ 1 is also obtained, and it is our aim to give a "more elementary" proof of that result, especially for the case 1 ≤ p < 2. Note that the proof we give below works also in the setting of the unit ball of C N , or other domains in C N .
Proof Suppose that h f is in the Schatten class S p . Since |V α f (z)| ≤ h f k z α , the case p ≥ 2 follows from the well known fact that h f k z α ∈ L p (D, dλ) is a necessary condition for h f being in S p if p ≥ 2. What is curious is that the case 1 ≤ p < 2 can be proved in a similar way. Indeed, one has Since 1 ≤ p < 2, using Hölder's inequality and (2.1), we obtain and this, together with another use of Hölder's inequality, gives This proves the necessity in the case 1 ≤ p < 2.
with h analytic on D, this implies that h ∈ A p β with β = (2 + α) p − 2. Since any function in A p β satisfies the growth condition it follows that V α f ∈ C 0 (D) proving that hf is compact. Next we proceed to show that hf is in S p . Since hf = h V α f , it suffices to prove that h ϕ belongs to S p whenever ϕ ∈ L p (D, dλ). To see this, it is enough to prove that n | h ϕ e n , f n α | p < ∞ for all orthonormal sets {e n } and { f n } of A 2 α . But notice that, by Fubini's theorem , and f 2 has the property that for any (or some) r > 0 the function G r defined by is bounded on D.
The case −1 < α < 1/( p − 1) was proved by Luecking in [10]. Also observe that for p = 2, condition (b) in the previous theorem coincides with condition (b) in Theorem 2, since H f k z α = f • ϕ z − P α ( f • ϕ z ) α , where ϕ z (w) = z−w 1−zw . The proof of Theorem 6 follows the same argument as in Theorem 2. As before, only the implication (d) implies (a) must be proved, since the others implications are well known. If (d) holds and g ∈ H ∞ , then using Proposition 2.4 there is a solution u of the equation ∂u = g∂ f 1 with u p,α ≤ C g p,α . Since any such solution must be of the form u = f 1 g + h for some h ∈ A p α , the boundedness of the Bergman projection P α : L p (D, d A α ) → A p α for p > 1 gives This shows that H f 1 is bounded, and the proof of the boundedness of H f 2 follows the same lines as in Theorem 2. Similarly, one can obtain the corresponding result for compactness of the Hankel operator.

The Two-Sided Ideal Problem
One can also consider a two-sided ideal problem, namely, for a function f ∈ L 2 (D, d A α ) describe, in terms of properties of f , the simultaneous membership of H f and Hf in S p (this is equivalent to the membership of H f in S p when f is real valued). In the case that p ≥ 2, Zhu obtained in [15] For the unweighted case α = 0, in [12] Xia proved that the same condition M O( f ) ∈ L p (D, dλ) still describes the simultaneous membership of H f and Hf in S p when 1 < p < 2. Note that for 0 < p ≤ 2/(2 + α), the condition M O α ( f ) ∈ L p (dλ) implies f being constant, so the natural conjecture for the weighted case is that the condition M O α ( f ) ∈ L p (dλ) will also be the correct condition for the case 2/(2 + α) < p < 2.
For this case one can use the method of J. Xia, but since it uses the duality between the S p classes, it can only be used to obtain results for p ≥ 1, and this will still give the gap 2/(2 + α) < p < 1 for the case α > 0. Therefore, it seems that new techniques are needed here.