Besov-Schatten Spaces

Abstract

We introduce the Besov-Schatten spaces , a matrix version af analytic Besov space, and we compute the dual of this space showing that it coincides with the matricial Bloch space introduced previously in Popa (2007). Finally we compute the space of all Schur multipliers on .

1. Introduction

Analytic Besov spaces first found its direct application in operator theory in Peller’s paper [1]. A comprehensive account of the theory of Besov spaces is given in Peetre’s book [2]. In what follows we consider the Besov-Schatten spaces in the framework of matrices, for example, infinite matrix-valued functions. The extension to the matriceal framework is based on the fact that there is a natural correspondence between Toeplitz matrices and formal series associated to -periodic functions (see, e.g., [3–6]). We use the powerful device Schur multipliers and its characterizations in the case of Toeplitz matrices to prove some of the main results.

The Schur product (or Hadamard product) of matrices and is defined as the matrix whose entries are the products of the entries of and : If and are two Banach spaces of matrices we define Schur multipliers from to as the space equipped with the natural norm In the case , where is the space of all linear and bounded operators on , the space will be denoted and a matrix will be called Schur multiplier. We mention here an important result due to Bennett [7], which will be often used in this paper.

Theorem 1.1. The Toeplitz matrix , where is a sequence of complex numbers, is a Schur multiplier if and only if there exists a bounded and complex Borel measure on (the circle group) with Moreover, one then has that

We will denote by , , the Schatten class operators (see, e.g., [8]). Let us summarize briefly some well-known properties of classes which will be very often used in what follows. If , then , where and . Next, interpolating between the classes , we can easily see that if (see, e.g., [9]). We will denote by , the -diagonal matrix associated to (see [4]). For an infinite matrix and an integer we denote by the matrix whose entries are given by In what follows we will recall some definitions from [10] (see also [11]), which we will use in this paper. We consider on the interval the Lebesgue measurable infinite matrix-valued functions . These functions may be regarded as infinite matrix-valued functions defined on the unit disc using the correspondence where is the -diagonal of the matrix , the preceding sum is a formal one, and belongs to the torus . This matrix is called analytic matrix if there exists an upper triangular infinite matrix such that, for all , we have , for all . In what follows we identify the analytic matrices with their corresponding upper triangular matrices and we call them also analytic matrices.

We also recall the definition of the matriceal Bloch space and the so-called little Bloch space of matrices (see [11]). The matriceal Bloch space is the space of all analytic matrices with , , such that where is the usual operator norm of the matrix on the sequence space and.

The space is the space of all upper triangular infinite matrices such that , where is the Toeplitz matrix associated with the Cauchy kernel , for .

An important tool in this paper is the Bergman projection. It is known (see, e.g., [10]) that for all strong measurable -valued functions defined on with and for all we have that

Now we consider a modified version of Bergman projection.

Let . Then

We remark that, for , it follows that .

We recall now a lemma from [11] that we will use in the following.

Lemma 1.2. Let , that is,
Then is an isomorphic embedding of in , where is the space of all continuous -valued functions on such that in the norm of .

The paper is organized as follows. In Section 2 we give a characterization of matrices in the Besov-Schatten space using the Bergman projection. The main result in Section 3 is a new duality result (see Theorem 3.2).

2. Besov-Schatten Spaces

Now we introduce a new space of matrices the so-called Besov-Schatten space.

Definition 2.1. Let and a positive measure on given by The Besov-Schatten matrix space is defined to be the space of all upper triangular infinite matrices such that

On we introduce the norm

We introduce the notation for the space of all strongly measurable functions defined on the measurable space with values such that

We need the following interesting lemma in what follows (see [8, page 53]).

Lemma 2.2. Let , is real, , and Then,(1)if , then is bounded in ;(2)if , then (3)if , then

The next theorem expresses a natural relation between the Bergman projection and the Besov-Schatten spaces. More precisely our main result of this section is the following equivalence theorem.

Theorem 2.3. Let and be an upper triangular matrix such that the -valued function is continuous on for some . Then the following assertions are equivalent:(1);(2);(3), where is the Bergman projection.

Proof. It is obvious that (1) is equivalent to (2). We observe that the Bergman projection may be described as follows: where . Then for all , and all .
It follows that each matriceal polynomial is in for all .
Suppose that is an upper triangular matrix with for all . We write where .
If , then we have that is in and moreover that .
Indeed, for , is a continuous function and, therefore Consequently .
Moreover since Thus we have proved that (2) implies (3).
It remains to prove that (3) implies (2). Suppose that (3) holds, and let for some . Then we have that Using Fubini’s theorem and Lemma 2.2 we obtain that
Consequently, and this proves that (3) implies (2) in the case . The proof in the case is similar to the classical case of functions (see, e.g., [8, Theorem  5.3.3.]). Let be the Toeplitz matrix with Since is a Schur multiplier with and , we get that From Schur’s theorem (see, e.g., [8]) it follows that is bounded on which in its turn implies that for . Thus also the implication (3)⇒(2) is proved and the proof is complete.

3. The Dual of Besov-Schatten Spaces

Our aim in this section is to characterize the Banach dual spaces of Besov-Schatten spaces.

First we prove the following lemma of independent interest.

Lemma 3.1. Let , that is,
Then is an embedding from into for all , if is equipped with the quotient norm.

Proof. Suppose that and with . Since it is easy to see that on . Therefore for all and .
We will now prove that is a bounded operator on. We first prove this fact for . By Fubini’s theorem we have that where means the Toeplitz matrix given by Since the Toeplitz matrix is a Schur multiplier with then, according to Lemma 2.2, it follows that Consequently is bounded on . For we have that where is a Toeplitz matrix and is a Schur multiplier, therefore From Schur’s theorem, (see, e.g., [8]) we obtain that is bounded on . Hence is bounded on , , and there is a constant such that for all . Taking the infimum over , we get that Thus is bounded.
On the other hand, since and on we get easily that for all . Thus and hence is an embedding. The proof is complete.

We denote by the closed Banach subspace of consisting of all upper triangular matrices whose diagonals are compact operators. Now we can formulate and prove the duality of Besov-Schatten spaces.

Theorem 3.2. Under the pairing One has the following dualities:(1) if and ;(2) and .

Proof. Since is an embedding from into for all , Hölder’s inequality shows that for and .
Suppose that is a bounded linear functional on the Besov-Schatten space with . Then extends to a bounded linear functional on . Thus there exists such that and In particular, if with , then Let . Then and it is easy to check that with . This proves the duality for .
It remains to prove the duality .
Let us assume that is a bounded linear functional on . Then we will prove that there is a matrix from such that for from a dense subset of . By Lemma 1.2 it follows that is an isomorphic embedding. Thus is a closed subspace in and is a bounded linear functional on , where is the subset in whose elements are -valued functions. By the Hahn-Banach theorem can be extended to a bounded linear functional on .
Let denote this functional. It follows that and, thus, is a bilinear integral map, that is, there is a bounded Borel measure on , where is the unit ball of the space with the topology , such that for every and .
Thus, for the matrix , identified with the analytic matrix , we have that On the other hand, we wish to have that Therefore, letting , denote the matrix having 1 as the single nonzero entry on the th-row and the th-column, for and , we have that Then, it yields that Consequently, and we get the relation (3.18) by using the fact that the set of all matrices is dense in .