© Hindawi Publishing Corp. TOEPLITZ OPERATORS WITH BMO SYMBOLS AND THE BEREZIN TRANSFORM

We prove that the boundedness and compactness of the Toeplitz operator on the Bergman space with a BMO1 symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends the known results in the cases when the symbol is either a positive L1-function or an L∞ function.


Introduction.
Toeplitz operators are one of the most widely studied classes of concrete operators.The study of their behavior on the Hardy and Bergman spaces has generated an extensive list of results in the operator theory and in the theory of function spaces.One of the latest approaches in this area is the use of the Berezin transform as a determining factor of the behaviour of the Toeplitz operator (see [1,2,6,8,10]).This method is motivated by its connections with quantum physics and noncommutative geometry.
We start with a few of the basic definitions.For more details and references, see [3,9].
The Bergman space L 2 a (D) is the subspace of L 2 (D) consisting of functions that are analytic on the unit disk D. Let P be the Bergman projection, that is, the projection form L 2 (D) onto L 2 a (D) defined by 1 − zω 2 dm(ω), (1.1) where dm denotes the normalized Lebesque area measure of D. For a function f in L 1 (D), the Toeplitz operator T f on L 2 a (D) is defined by Since the Bergman projection kernel function P can be extended to L 1 (D), the operator T f is well defined on H ∞ , the space of bounded analytic functions on D. Hence, T f is always densely defined on L 2 a (D).Since P is not bounded on L 1 (D), T f can be unbounded in general.
For z in D, the Bergman kernel function K z and the normalized Bergman kernel function k z are functions in L 2 a (D), defined by (1.3) We have that, for any g in L 2 a (D), g, K z = g(z).
(1.4) Also, k z is in H ∞ and k z 2 = 1, where •, • denotes the inner product in L 2 (D) and • 2 denotes the L 2 (D) norm.
For an operator A on L 2 a (D) that is well defined on H ∞ , the Berezin transform of A is the function A on D defined by (1.5) If A is bounded, then A is a bounded function.Since the kernels k z converge weakly to zero as z approaches the unit circle ∂D, we have that if A is compact, then A(z) → 0 as z → ∂D.The converse (in both cases) is not necessarily true and we will mention some counterexamples later on.For f in L 1 (D), we define the Berezin transform of f to be the function T f , that is, 2 dm(ω). (1.6) Our main result states that for f in the space BMO 1 (D) (to be defined later), T f is bounded if and only if f is bounded and T f is compact if and only if f (z) → 0 as z → ∂D.The same result has been proven for positive L 1 (D) symbols f by Luecking and Zhu in [5,8], and for L ∞ (D) symbols f by Axler and Zheng in [1].We will see that both of these classes of symbols are contained in BMO 1 (D) and so our result covers the above two cases.
We mention few more properties of the Berezin transform function (more details can be found in [1,3,9]).
(1) The map A → A is one to one. ( More precisely, A(z) is real analytic on D with a power series expansion (1.7) (3) For f in L 1 (D), f is harmonic on D if and only if f = f .While (1) and ( 2) are fairly easy to obtain, property (3) is a very deep result that was an open conjecture for a number of years, until it was proved independently by Ahern, Flores, and Rudin in 1993 and by English in 1994.For detailed references, see [3].
There are several examples in the literature of noncompact operators with Berezin transform vanishing at the boundary (see, e.g., [1]).We mention one of them and then give an example of an unbounded operator with a bounded Berezin transform.Both of the operators will be radial operators, that is, operators that are diagonal with respect to the standard basis {e n } of L 2 a (D), where e n (z) = √ n + 1z n (see [10] for more details on radial operators and their Berezin transform).
Example 1.1 (see [1, page 392]).Let A be the diagonal operator on L 2 a (D) defined by where m ∈ N.
Example 1.2.Let A be the diagonal operator on L 2 a (D) defined by where m ∈ N. The operator A is unbounded on L 2 a (D).We will show that (1.11) Thus A(z) ≤ 6|z| 2 ≤ 6 for all z in D.
Note that none of the operators in Examples 1.1 and 1.2 is a Toeplitz operator with symbol in L 1 (D) (see [10] for details).
We will define the BMO 1 (D) spaces for p ≥ 1 in Section 2 and we will describe some of the properties of the functions belonging to these spaces.The proof of our main result will be presented in Section 3.
Throughout the paper, we will use the letter c to denote a generic positive constant that can change its value at each occurrence.
2. BMO p spaces.Let f ∈ L 1 (D) and let ψ z denote the disk automorphism defined by For p ≥ 1, we say that f belongs to BMO p (D) whenever where • p denotes the L p (D) norm, and f is the Berezin transform of f .We define ( Note that • BMO p does not distinguish constants, while | • | p is a norm in BMO p (D). BMO p (D) spaces were introduced for p = 2 by Békollé et al. (see [2]) and for general p ≥ 1 (and general pseudoconvex domains) by Li and Luecking in [4].There are several equivalent norms on BMO p (D) that appear in the literature.We will mention another definition stemming from the traditional approach to the BMO spaces on the unit circle.This definition gives a geometric view of the BMO p (D) spaces by explicitly using the Bergman metric.
For f in L 1 (D), the average of f over D(z) is defined by Using properties of the Bergman metric and results from [4], it follows that f BMO p is finite if and only if Thus, functions in BMO p (D) have bounded mean oscillation in the Bergman metric.Note that any other choice for the radius of the hyperbolic disk D(z) gives the same set of functions.Since BMO p (D) functions are locally in L p (D), the spaces are different for different p.It is not hard to see that Details regarding (2.6) could be found in [4,7,9].Since the space BMO 1 (D) is the largest among the BMO p (D) spaces for p ≥ 1, from now on we will be mainly interested in functions belonging to this class.We will also drop the reference to the unit disk and simply write BMO p instead of BMO p (D).
The next proposition says more about the Berezin transform of BMO 1 functions.Similar properties have been proven about BMO 2 functions (see [2]).
where we have used the change of variable ω = ψ z (v) in the third line of (2.7). Since (b) We have to show that there exists a constant c > 0 such that, for every z, ω ∈ D, (2.8) Békollé et al. have established in [2] that the same property for f is true in the case when f belongs to BMO 2 .We will explain the main idea of the proof and the part where our proof differs from that in [2].For z, ω ∈ D, let α(t) denote the geodesic from z = α(0) to ω = α(1) in the Bergman metric, and let s = s(t) denote the arc length of α(t) in the Bergman metric.Since The following inequality can be found in [3, page 48] and is a part of the above-mentioned proof in [2] that we use here: (2.10) Since α (t) and since we get that (2.13) Hence, and the constant c can be chosen to be equal to 4 (2.16) Using the fact that lim h→0 (β(z + h, z)/|h|) = 1/(1 −|z| 2 ) and part (b), we get that (2.17) Similarly, (2.18) (2.21) As a consequence of the results in Propositions 2.1 and 2.2, we get the following corollary.
Proof.(a) Using the fact that for every f in L 1 (D) we have it follows from Proposition 2.1(a) that whenever f is in BMO 1 and f is bounded, we have that |f | is also bounded.Since |f | ≥ 0, by results from [5,8], |f | being bounded implies that T |f | is bounded.Then it is not hard to see that T f has to be bounded too.(b) Take f 1 = f and f 2 = f − f .Then the rest follows from Proposition 2.1(c) and (d).
(c) For In case f is bounded, using Proposition 2.2(a), we get that f belongs to BMO 1 .

Proof of the main theorem.
Our main theorem (Theorem 3.1) expands the class of functions f for which it is known that f (z) → 0, as z → ∂D implies that T f is compact.It includes L ∞ functions and positive L 1 functions with bounded Berezin transform, and so the theorem is an extension of the results of Axler and Zheng (see [1]) and Luecking and Zhu (see [5,8]).
Before we proceed with the proof, we state two lemmas that contain some of the more technical parts used in the proof.Lemma 3.2.Let f be in L 1 (D) and let T f be bounded on L 2 a (D).Then for every z in D, the following is true: Proof.(a) We have the following equalities: where we have used the change of variable ω = ψ z (v), and the fact that Using the equations (which can be checked directly from the definitions of the functions involved), we get that by the change of the variable ω = ψ z (u).Using the fact that we can continue with the following equations: where we have used the change of the variable v = ψ z (ω) in the fourth equation.Thus T f k z (c) It follows from the definition of BMO 1 that, whenever f belongs to BMO 1 , f • ψ z also belongs to BMO 1 , for all z in D. Furthermore, (where we have used that is bounded in z.
The proof of the theorem will be done in several steps.The steps follow the standard idea of finding a sequence of compact operators that converges to the given operator.To establish the convergence, we will use the Schur's test.The same approach has also been implemented in [1,6].The core of our extension is contained in the first step of the proof.

Proof of Theorem 3.1
Step 1.Let f ∈ BMO 1 and let f be bounded.Then (3.10)
The Bloch spaces B is defined by For g, being a function in the Bloch space B, let (3.12) It has been proven by Li and Luecking in [4] that the Bergman projection P is a bounded operator from BMO 1 into B, for all p ≥ 1. Since, for f in BMO 1 we have that f • ψ z is in BMO 1 for all z in D, we get that P (f (3.13) So, for f in BMO 1 and f being bounded, we get that (3.14) Step 2. Let T f be bounded on L 2 a (D) and let f (z) → 0 as z → ∂D.Then T f •ψz 1 → 0 weakly as z → ∂D.

Proof of
Step 2 (see [1, page 396]).The proof uses the explicit doubleseries form of the Berezin transform of an operator in L 2 a (D).We mention that, since the specific nature of the Toeplitz operator is not used in the proof, a more general statement is true.
Let A be a bounded operator on L 2 a (D) and let U z be the unitary operator on L 2 a (D), defined by
Proof of Step 3. The method of the proof is similar to a part of the proof in [1].For the sake of completeness, we provide the details.
By Step 2, T f •ψz 1 → 0 weakly and so it converges uniformly to zero on compact subsets of D, such as r D, for 0 ≤ r < 1.Since we also need to estimate the first integral of the last line.We will do that by using the Cauchy-Schwartz inequality and the result of Step 1: (3.17) We can make the integral over D \ r D as small as we wish, independently of z, by choosing r close enough to 1.Then, for the same r , take z close enough to ∂D such that the integral over r D is also as small as we wish.
Step 5. Let f ∈ BMO 1 and let f (z) → 0 as z → ∂D.For 0 < r < 1, let T f r be the operator from L 2 a (D) into L 2 (D), defined by where M χ r D is the multiplication operator on L 2 a (D) with χ r D being the characteristic function of r D. Let T f denote the operator T f as an operator from Proof of Step 5.It is well known that the operator M χ r D is compact on L 2 (D) since χ r D (z) = 0 for |z| ≥ r .Thus, T f r is also compact, as a product of a compact and a bounded operator.
For g in L 2 a (D), we have that . By Schur's test, whenever there exist a positive measurable function h on D and constants c 1 and c 2 such that We will show that the Schur's test works with constants We have that (3.23) which by Lemma 3.2(a) equals By the change of variable ψ z (u) = v and by Hölder's inequality with p as above, we get that To get the last equality, we have used the equation Using it one more time and then applying Lemma 3.3 in the last inequality, we get that (3.27) We get the first inequality of the Schur's test in a similar way, noting that By the same argument as above, the right-hand side is less than or equal to

Comments and further generalizations.
A general problem that motivated the results of this paper is to determine the class of operators A on the Bergman space for which A(z) → 0, as z→ ∂D, implies that A is compact.This class of C ∞ (D) functions, vanishing on the boundary, has to be an ideal of the noncommutative algebra of Berezin transform functions on D.
Our result states that Toeplitz operators with BMO 1 symbols belong to this ideal.The method of the proof actually yields a stronger result that we state below as Theorem 4.2.The following lemma considers the main technical generalization.
Proof.Since q < p, it follows that sup z∈D T f •ψz 1 q < ∞.Let s = p/q and let t be such that 1/t + 1/s = 1.Then By the comment in the proof of Step 2 and since f (z) → 0 as z→ ∂D, it follows that T f •ψz 1 → 0 weakly as z → ∂D, and so I 1 (z) → 0 as z → ∂D.But which could be made arbitrarily small independently of z by taking r close enough to 1. Thus T f •ψz 1 q → 0, as z → ∂D.
Theorem 4.2.Let f be in L 1 (D), let T f be bounded in L 2 a (D), and suppose that there exists p > 3 such that Proof.By Lemma 4.1, there exists q > 3 such that sup z∈D T f •ψz 1 q < ∞ and T f •ψz 1 q → 0 as z → ∂D.The same holds for T f •ψz 1 q .Closer inspection of the proof of Step 5 shows that this is sufficient to show that lim r →1 T f r − T f = 0 (where T f r and T f are defined in Step 5).Hence T f is compact on L 2 a (D).
It is not known if for every bounded Toeplitz operator T f , f (z) → 0 as z → ∂D, guarantees that T f is compact.Note that T f being bounded on L 2 a (D) implies that sup z∈D T f •ψz 1 2 < ∞.
We ask the following question.
Question.For f in L 1 (D) and T f being bounded on L 2 a (D), and such that sup z∈D T f •ψz 1 p < ∞ and sup z∈D T f •ψz 1 p < ∞ for some p > 2, does it follow that f (z) → 0 as, z → ∂D, implies that T f is compact?

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .29) Thus, by Schur's test, T f − T f r 2 ≤ c 1 c 2 , where c 1 does not depend on r , and by Step 4, c 2 → 0 as r → 1.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation