( m , λ )-Berezin Transform on the Weighted Bergman Spaces over the Polydisk

L λ (D). As is well knownA λ (D) forms a closed subspace of L λ (D) and has the structure of reproducing kernel Hilbert space. We denote by B λ the Bergman projection of L λ (D) onto A λ (D). In case of λ = 0, A 0 (D) is the unweighted Bergman space denoted by A(D). Given an essentially bounded measurable function a ∈ L(D), we write T a for the Toeplitz operator with the symbol a, which acts on A λ (D) as T a f = B λ (af). That is, the Toeplitz operator is defined as the compression of a multiplication operator on


Introduction
Let D be the unit disk in C and   () =   (1−|| 2 )  () be a positive standard weighted probability measure on D, where the weighted parameter fulfills  > −1 and the normalized constant   = +1.For a fixed positive integer , the polydisk D  is the Cartesian product of  copies of D and is the normalized weighted Lebesgue volume measure on the polydisk D  .The Bergman space A 2  (D  ) = A 2  (D  ,   ) is the set of all analytic functions on D  in  2  (D  ,   ) =  2   (D  ).As is well known A 2  (D  ) forms a closed subspace of  2   (D  ) and has the structure of reproducing kernel Hilbert space.We denote by   the Bergman projection of  2  (D  ) onto A 2  (D  ).In case of  = 0, A 2 0 (D  ) is the unweighted Bergman space denoted by A 2 (D  ).Given an essentially bounded measurable function  ∈  ∞ (D  ), we write   for the Toeplitz operator with the symbol , which acts on A 2  (D  ) as    =   ().That is, the Toeplitz operator is defined as the compression of a multiplication operator on  2   (D  ) onto the Bergman space.The Toeplitz algebra T( ∞ ) is the closed subalgebra of L(A 2  ) generated by {  :  ∈  ∞ (D  )}, where L(A 2  ) denotes the algebra of all bounded linear operators on A 2  (D  ).Due to their simple structure Toeplitz operators form an important, tractable, and intensively studied subclass in the algebra L(A 2  ) of all bounded linear operators on A 2  (D  ).The natural question is whether the Toeplitz algebra is dense in the algebra of all bounded linear operators on the Bergman space.On unweighted Bergman space over the unit disk and even more general domain in C, it is proved in [1] that the Toeplitz algebra is dense in the algebra of all bounded linear operators in the sense of strong operator topology (SOT).In general, it is not true if the SOT is replaced by the norm topology.
Nam and Zheng give a criterion for bounded operators approximated by Toeplitz operators on A 2 (D  ).Since the Berezin transform is a useful tool to study operators on any reproducing kernel Hilbert space, the -Berezin transform for any bounded linear operators acting on A 2 (D  ) was defined in [2].The operator  ∈ L( 2  ) can be approximated in the norm by Toeplitz operators on the unit ball (see [3]) by using the -Berezin transform.In [4], the (, )-Berezin transform for complex-valued regular measures on the weighted -Bergman space over the unit ball was defined and studied.Using it, they show that every  ∈ T( ∞ ) can be approximated by certain localized operators and introduce a way to connect the behavior of these localized operators with the Berezin transform.The (, )-Berezin transform for general bounded operators acting on the weighted Bergman space A 2  (B  ) was defined in [5] and the authors establish various results on norm approximations via the (, )-Berezin transform and describe conditions under which a bounded linear operator  can be approximated in norm by Toeplitz operators whose symbols are bounded functions.
In this paper, we will define and study the (, )-Berezin transform for general bounded operators acting on the weighted Bergman space A 2  (D  ) in the third section.The (, )-Berezin transform of a Toeplitz operator   acting on A 2  (D  ) coincides with (0,  + )-Berezin transform for   considered on the weighted Bergman space A 2 + (D  ).We will show that the (, )-Berezin transforms are commuting with each other.In Section 4, we will establish various results on norm approximation by the (, )-Berezin transform.More precisely, we describe how to approximate a bounded linear operator  on A 2  (D  ) in norm by Toeplitz operators whose symbols are bounded functions which are given as the (, )-Berezin transform of the initial operator  under some conditions.We would like to point out that these results generalize ideas and theorems in [2] to the case of operators acting on A 2  (D  ).
For a fixed  ∈ D  we define an automorphism on the algebra L(A 2  ) of all bounded operator on The principle difference between the unit ball B  and the polydisk D  is that the later domain is reducible, which involves the tensor product structure of various objects introduced and studied in the paper.In particular, (D,    (  )).Therefore, for the orthonormal basis of A 2  (D  ) and the reproducing kernel in A 2  (D  ), we have The unitary operator   on A 2  (D  ) can be written by Throughout the paper and as a convention we will denote by (, ) a positive constant depending only on  and  and appearing in various estimates and whose value may change at each occurrence.

The (𝑚, 𝜆)-Berezin Transform
Recall that -Berezin transform for unweighted Bergman space over the unit disk and over the unit polydisk was defined in [6] and [2], respectively.We will follow the recipe in [2] and first introduce some notation.Put so that, for ,  ∈ D  , we know For  ∈ D  ,  = ( 1 , . . .,   ) ∈ R  , and a positive integer , ).A generalization of the concept of -Berezin transform to an arbitrary bounded operator on the Bergman space A 2 (D  ) requires a modification of the definition in [2].Definition 1.We define the (, )-Berezin transform of  ∈ L(A 2  ) by It is easy to see that the following pointwise estimate where the constant (, , ) > 0 is independent of  ∈ D  ; that is,  ,  is a bounded function on D  with ‖ , ‖ ∞ ≤ (, , )‖‖.
In [5], the (, )-Berezin transform of  ∈ L(A 2  (D)) is defined by ( , )() = (( +  + 1)/( + 1)) ∑  =0 (−1)  (   ) ⟨    ,   ⟩, for the case of the unit disk B 1 = D. From the point of view of the tensor product structure, given an elementary tensor ), its (, )-Berezin transform for  = ( 1 , . . .,   ) obviously and naturally has to be defined as Unfortunately, the set of those tensor product operators is not a linear space; that is, for any operator  ∈ L(A 2  (D  )),  cannot be written in the form of the tensor product operators.Therefore, we define for any operator  ∈ L(A 2  (D  )) with (10), and this coincides with Definition 1.If  can be the tensor product form, this definition is the tensor product of (,   )-Berezin transform for the case of  = 1.
In the first inequality we use that (⋅, ⋅) is invariant under the automorphisms   and by the Lipschitz continuity of  0, .Now we estimate the second integral above.
It is clear that the right-hand side converges to zero as  → ∞.

Toeplitz Operators to Approximate the Bounded Operators
In this section we will give a criterion for an operator approximated by Toeplitz operators with symbol equal to their (, )-Berezin transforms.From Proposition 1.4.10 in [7] there exists Lemma 3.1 in [2].
Proof.Given  ∈ D  , the equality (37) which holds for all ,  ∈ D  .