Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 693251 | https://doi.org/10.1155/2012/693251

A. N. Marcoci, L. G. Marcoci, L. E. Persson, "Besov-Schatten Spaces", Journal of Function Spaces, vol. 2012, Article ID 693251, 13 pages, 2012. https://doi.org/10.1155/2012/693251

Besov-Schatten Spaces

Academic Editor: Nicolae Popa
Received22 Mar 2011
Accepted09 Apr 2011
Published15 Jan 2012

Abstract

We introduce the Besov-Schatten spaces 𝐵𝑝(2), 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 𝐵1(2).

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 2𝜋-periodic functions (see, e.g., [36]). 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 𝐴=(𝑎𝑗𝑘)𝑗,𝑘0 and 𝐵=(𝑏𝑗𝑘)𝑗,𝑘0 is defined as the matrix 𝐴𝐵 whose entries are the products of the entries of 𝐴 and 𝐵: 𝑎𝐴𝐵=𝑗𝑘𝑏𝑗𝑘𝑗,𝑘0.(1.1) If 𝑋 and 𝑌 are two Banach spaces of matrices we define Schur multipliers from 𝑋 to 𝑌 as the space 𝑀(𝑋,𝑌)={𝑀𝑀𝐴𝑌forevery𝐴𝑋},(1.2) equipped with the natural norm 𝑀=sup𝐴𝑋1𝑀𝐴𝑌.(1.3) In the case 𝑋=𝑌=𝐵(2), where 𝐵(2) is the space of all linear and bounded operators on 2, the space 𝑀(𝐵(2),𝐵(2)) will be denoted 𝑀(2) and a matrix 𝐴𝑀(2) 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 𝑀=(𝑐𝑗𝑘)𝑗,𝑘, where (𝑐𝑛)𝑛 is a sequence of complex numbers, is a Schur multiplier if and only if there exists a bounded and complex Borel measure 𝜇 on (the circle group) 𝕋 with 𝜇(𝑛)=𝑐𝑛,for𝑛=0,±1,±2,.(1.4) Moreover, one then has that 𝑀=𝜇.(1.5)

We will denote by 𝐶𝑝, 0<𝑝<, the Schatten class operators (see, e.g., [8]). Let us summarize briefly some well-known properties of classes 𝑀(𝐶𝑝) which will be very often used in what follows. If 1<𝑝<, then 𝑀(𝐶𝑝)=𝑀(𝐶𝑝), where 1/𝑝+1/𝑝=1 and 𝑀(2)=𝑀(𝐶1). Next, interpolating between the classes 𝐶𝑝, we can easily see that 𝑀(𝐶𝑝1)𝑀(𝐶𝑝2) if 0<𝑝1𝑝22 (see, e.g., [9]). We will denote by 𝐴𝑘, the 𝑘th-diagonal matrix associated to 𝐴 (see [4]). For an infinite matrix 𝐴=(𝑎𝑖𝑗) and an integer 𝑘 we denote by 𝐴𝑘 the matrix whose entries 𝑎𝑖𝑗 are given by 𝑎𝑖𝑗=𝑎𝑖𝑗if𝑗𝑖=𝑘,0otherwise.(1.6) 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 [0,1) the Lebesgue measurable infinite matrix-valued functions 𝐴(𝑟). These functions may be regarded as infinite matrix-valued functions defined on the unit disc 𝐷 using the correspondence 𝐴(𝑟)𝑓𝐴(𝑟,𝑡)=𝑘=𝐴𝑘(𝑟)𝑒𝑖𝑘𝑡,(1.7) where 𝐴𝑘(𝑟) is the 𝑘th-diagonal of the matrix 𝐴(𝑟), the preceding sum is a formal one, and 𝑡 belongs to the torus 𝕋. This matrix 𝐴(𝑟) is called analytic matrix if there exists an upper triangular infinite matrix 𝐴 such that, for all 𝑟[0,1), we have 𝐴𝑘(𝑟)=𝐴𝑘𝑟𝑘, for all 𝑘. In what follows we identify the analytic matrices 𝐴(𝑟) with their corresponding upper triangular matrices 𝐴 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 (𝐷,2) is the space of all analytic matrices 𝐴 with 𝐴(𝑟)𝐵(2), 0𝑟<1, such that 𝐴(𝐷,2)=sup0𝑟<11𝑟2𝐴(𝑟)𝐵(2)+𝐴0𝐵(2)<,(1.8) where 𝐵(2) is the usual operator norm of the matrix 𝐴 on the sequence space 2 and𝐴(𝑟)=𝑘=0𝐴𝑘𝑘𝑟𝑘1.

The space 0(𝐷,2) is the space of all upper triangular infinite matrices 𝐴 such that lim𝑟1(1𝑟2)(𝐴𝐶(𝑟))𝐵(2)=0, where 𝐶(𝑟) is the Toeplitz matrix associated with the Cauchy kernel 1/(1𝑟), for 0𝑟<1.

An important tool in this paper is the Bergman projection. It is known (see, e.g., [10]) that for all strong measurable 𝐶𝑝-valued functions 𝑟𝐴(𝑟) defined on [0,1) with 10𝐴(𝑟)𝑝𝐶𝑝2𝑟𝑑𝑟< and for all 𝑖,𝑗 we have that []𝑃(𝐴())(𝑟)(𝑖,𝑗)=2(𝑗𝑖+1)𝑟𝑗𝑖10𝑎𝑖𝑗(𝑠)𝑠𝑗𝑖+1𝑑𝑠,if𝑖𝑗,0,otherwise.(1.9)

Now we consider a modified version of Bergman projection.

Let 𝛼>1. Then𝑃𝛼𝐴()(𝑟)=(𝛼+1)Γ(𝑗𝑖+2+𝛼)𝑟(𝑗𝑖)!Γ(𝛼+2)𝑗𝑖210𝑎𝑖𝑗(𝑠)𝑠𝑗𝑖+11𝑠2𝛼𝑑𝑠if𝑗𝑖0if𝑗<𝑖.(1.10)

We remark that, for 𝛼=0, it follows that 𝑃𝛼=𝑃.

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

Lemma 1.2. Let 𝑉=(𝑃2), that is, 𝑃2𝐴()=(𝑟)(𝑖,𝑗)(𝑗𝑖+3)(𝑗𝑖+2)(𝑗𝑖+1)2𝑟𝑗𝑖1𝑟2210𝑎𝑖𝑗(𝑠)𝑠𝑗𝑖(2𝑠𝑑𝑠)if𝑗𝑖00otherwise.(1.11)
Then 𝑉 is an isomorphic embedding of 0(𝐷,2) in 𝒞0(𝐷,2), where 𝒞0(𝐷,2) is the space of all continuous 𝐵(2)-valued functions 𝐵(𝑟) on [0,1) such that lim𝑟1𝐵(𝑟)=0 in the norm of 𝐵(2).

The paper is organized as follows. In Section 2 we give a characterization of matrices in the Besov-Schatten space 𝐵𝑝(2) 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 1𝑝< and a positive measure on [0,1) given by 𝑑𝜆(𝑟)=2𝑟𝑑𝑟1𝑟22.(2.1) The Besov-Schatten matrix space 𝐵𝑝(2) is defined to be the space of all upper triangular infinite matrices 𝐴 such that 𝐴𝐵𝑝(2)=101𝑟22𝑝𝐴(𝑟)𝑝𝐶𝑝𝑑𝜆(𝑟)1/𝑝<.(2.2)

On 𝐵𝑝(2) we introduce the norm 𝐴𝐴=0𝐶1+𝐴𝐵𝑝(2).(2.3)

We introduce the notation 𝐿𝑝(𝐷,𝑑𝜆,2) for the space of all strongly measurable functions 𝑟𝐴(𝑟) defined on the measurable space ([0,1),𝑑𝜆) with 𝐶𝑝 values such that 𝐴𝐿𝑝(𝐷,𝑑𝜆,2)=10𝐴(𝑟)𝑝𝐶𝑝𝑑𝜆(𝑟)1/𝑝<.(2.4)

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

Lemma 2.2. Let 𝑧𝐷, 𝑐 is real, 𝑡>1, and 𝐼𝑐,𝑡=𝐷1|𝑤|2𝑡||1𝑧𝑤||2+𝑡+𝑐𝑑𝐴(𝑤).(2.5) Then,(1)if 𝑐<0, then 𝐼𝑐,𝑡(𝑧) is bounded in 𝑧;(2)if 𝑐>0, then 𝐼𝑐,𝑡1(𝑧)1|𝑧|2𝑐(|𝑧|1);(2.6)(3)if 𝑐=0, then 𝐼0,𝑐1(𝑧)log1|𝑧|2(|𝑧|1).(2.7)

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 1𝑝< and 𝐴 be an upper triangular matrix such that the 𝐶𝑝-valued function 𝑟𝐴(𝑟) is continuous on [0,𝑟0) for some 1>𝑟0>0. Then the following assertions are equivalent:(1)𝐴𝐵𝑝(2);(2)(1𝑟2)2𝐴(𝑟)𝐿𝑝(𝐷,𝑑𝜆,2);(3)𝐴𝑃𝐿𝑝(𝐷,𝑑𝜆,2), where 𝑃 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: 𝑃(𝐴())=𝑘=0(𝑘+1)10[]𝐴(𝑠)𝑘𝑠𝑘(2𝑠𝑑𝑠),(2.8) where 𝐴()𝐿𝑝(𝐷,2). Then 𝑃1𝑟22𝐴𝑘𝑟𝑘(𝑠)=2𝑠𝑘𝐴𝑘,(𝑘+2)(𝑘+3)(2.9) for all 𝑘0, and all 𝐴𝑘𝐶𝑝.
It follows that each matriceal polynomial is in 𝑃𝐿𝑝(𝐷,𝑑𝜆,2) for all 1𝑝<.
Suppose that 𝐴 is an upper triangular matrix with 𝐴𝑘𝐶𝑝 for all 𝑘0. We write 𝐴=4𝑘=0𝐴𝑘+𝐴1,(2.10) where 𝐴1=𝑘=5𝐴𝑘.
If (1𝑟2)2𝐴(𝑟)𝐿𝑝(𝐷,𝑑𝜆,2), then we have that Φ(𝑟)=4𝑘=0(𝑘+2)(𝑘+3)21𝑟22𝐴𝑘𝑟𝑘+1𝑟22𝐴1(𝑟)2!𝑟2(2.11) is in 𝐿𝑝(𝐷,𝑑𝜆,2) and moreover that 𝐴=𝑃Φ.
Indeed, for 0<𝑟<𝑟0, 𝑟(𝐴1)(𝑟) is a continuous function and, therefore 𝑟00𝐴1(𝑠)𝑝𝐶𝑝𝑠2𝑝𝑑𝑠<.(2.12) Consequently Φ𝐿𝑝(𝐷,𝑑𝜆,2).
Moreover 𝐴=𝑃Φ since 𝑘=510𝑘(𝑘1)𝐴𝑘𝑠𝑘21𝑠22𝑠2(𝑘+1)𝑠𝑘+1=𝑑𝑠𝑘=5(𝑘1)𝑘(𝑘+1)𝐴𝑘10𝑠2𝑘31𝑠22𝑑𝑠=𝑘=5𝐴𝑘.(2.13) Thus we have proved that (2) implies (3).
It remains to prove that (3) implies (2). Suppose that (3) holds, and let 𝐴=𝑃Φ for some Φ()𝐿𝑝(𝐷,𝑑𝜆,2). Then we have that 1𝑟22𝐴(𝑟)=1𝑟2210Φ(𝑠)6𝑠2(1𝑟𝑠)42𝑠𝑑𝑠.(2.14) Using Fubini’s theorem and Lemma 2.2 we obtain that 101𝑟22𝐴(𝑟)𝐶12𝑟𝑑𝑟1𝑟221010Φ(𝑠)𝐶102𝜋6𝑠2𝑑𝜃||1𝑟𝑠𝑒𝑖𝜃||4=2𝑠𝑑𝑠2𝑟𝑑𝑟106𝑠2(Φ𝑠)𝐶11012𝜋02𝜋𝑑𝜃||1𝑟𝑠𝑒𝑖𝜃||42𝑟𝑑𝑟2𝑠𝑑𝑠10Φ(𝑠)𝐶112𝑠31𝑠22𝑑𝑠610Φ(𝑠)𝐶1𝑑𝜆(𝑠)<.(2.15)
Consequently, 𝐴𝐿1(𝐷,𝑑𝜆,2) and this proves that (3) implies (2) in the case 𝑝=1. The proof in the case 1<𝑝< is similar to the classical case of functions (see, e.g., [8, Theorem  5.3.3.]). Let 𝑇(𝑟𝑠)=((𝑡𝑖𝑗)(𝑟𝑠))𝑖,𝑗=1 be the Toeplitz matrix with 𝑡𝑖𝑗(𝑟𝑠)=𝑡𝑗𝑖𝑠(𝑟𝑠)=2(𝑟𝑠)𝑗𝑖(𝑗𝑖+3)(𝑗𝑖+2)(𝑗𝑖+1)if𝑗𝑖0otherwise.(2.16) Since 𝑇(𝑟𝑠) is a Schur multiplier with 𝑇(𝑟𝑠)𝑀(2)=𝑇(𝑟𝑠)𝐿1(𝕋)=6𝑠2/(1𝑟𝑠𝑒𝑖𝜃)4𝐿1(𝕋) and 𝑀(2)=𝑀(𝐶1)𝑀(𝐶𝑝), 1𝑝< we get that 1𝑟22𝐴(𝑟)𝐶𝑝=1𝑟2210𝜙(𝑠)6𝑠2(1𝑟𝑠)4(2𝑠𝑑𝑠)𝐶𝑝1𝑟2210𝜙(𝑠)𝐶𝑝6𝑠21𝑟𝑠𝑒𝑖𝜃4𝐿1(𝕋)=(2𝑠𝑑𝑠)1𝑟2210𝜙(𝑠)𝐶𝑝1𝑠226𝑠21𝑟𝑠𝑒𝑖𝜃4𝐿1(𝕋)𝑑𝜆(𝑠)=𝑆𝜙(𝑟).(2.17) From Schur’s theorem (see, e.g., [8]) it follows that 𝑆𝜙(𝑟) is bounded on 𝐿𝑝([0,1),𝑑𝜆) which in its turn implies that 1𝑟22𝐴(𝑟)𝐿𝑝𝐷,𝑑𝜆,2(2.18) for 1<𝑝<. 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 𝑉=(𝑃2), that is, []=𝑉(𝐴())(𝑟)(𝑖,𝑗)(𝑗𝑖+3)(𝑗𝑖+2)(𝑗𝑖+1)2𝑟𝑗𝑖1𝑟2210𝑎𝑖𝑗(𝑠)𝑠𝑗𝑖(2𝑠𝑑𝑠)if𝑗𝑖0,0otherwise.(3.1)
Then 𝑉 is an embedding from 𝐵𝑝(2) into 𝐿𝑝(𝐷,𝑑𝜆,2) for all 𝑝1, if 𝐵𝑝(2)=𝑃𝐿𝑝(𝐷,𝑑𝜆,2) is equipped with the quotient norm.

Proof. Suppose that 𝐴𝐵𝑝(2) and 𝐵()𝐿𝑝(𝐷,𝑑𝜆,2) with 𝐴=𝑃𝐵(). Since 𝑃(𝐵())(𝑟)(𝑖,𝑗)=2(𝑗𝑖+1)𝑟𝑗𝑖10𝑏𝑖𝑗(𝑠)𝑠𝑗𝑖+1𝑑𝑠if𝑗𝑖00otherwise,(3.2) it is easy to see that 𝑃𝑉=𝑃,𝑉𝑃=𝑉(3.3) on 𝐿𝑝(𝐷,𝑑𝜆,2). Therefore 𝑉(𝐴)=𝑉(𝐵()) for all 𝐴𝐵𝑝(2) and 𝐵()𝐿𝑝(𝐷,𝑑𝜆,2).
We will now prove that 𝑉 is a bounded operator on𝐿𝑝(𝐷,𝑑𝜆,2). We first prove this fact for 𝑝=1. By Fubini’s theorem we have that 𝑉(𝐴())𝐿1(𝐷,𝑑𝜆,2)=10[]𝑉(𝐴())𝐶1=𝑑𝜆(𝑟)10𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘1𝑟2210𝐴𝑘(𝑠)𝑠𝑘(2𝑠𝑑𝑠)𝐶1𝑑𝜆(𝑟)10𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘(1𝑟2)2𝐴𝑘(𝑠)𝑠𝑘+1𝐶1=(2𝑑𝑠)𝑑𝜆(𝑟)1010𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘𝐴𝑘(𝑠)𝑠𝑘1𝑠22𝐶1=(2𝑟𝑑𝑟)𝑑𝜆(𝑠)10𝐴(𝑠)𝐶(𝑟𝑠)𝐶1(2𝑟𝑑𝑟)𝑑𝜆(𝑠),(3.4) where 𝐶(𝑟𝑠)=(𝑐𝑖𝑗(𝑟𝑠))𝑖,𝑙=1 means the Toeplitz matrix given by 𝑐𝑖𝑗(𝑟𝑠)=𝑐𝑗𝑖=((𝑟𝑠)𝑟𝑠)𝑗𝑖𝑠21𝑠22(𝑗𝑖+3)(𝑗𝑖+2)(𝑗𝑖+1)2if𝑗𝑖,0otherwise.(3.5) Since the Toeplitz matrix 𝐶(𝑟𝑠) is a Schur multiplier with 𝐶(𝑟𝑠)𝑀(2)=6𝑠21𝑠221𝑟𝑠𝑒𝑖𝜃4𝐿1(𝕋),(3.6) then, according to Lemma 2.2, it follows that 10𝐴(𝑠)𝐶(𝑟𝑠)𝐶1(2𝑟𝑑𝑟)𝑑𝜆(𝑠)10𝐴(𝑠)𝐶110𝐶(𝑟𝑠)𝑀(2)(2𝑟𝑑𝑟)𝑑𝜆(𝑠)10𝐴(𝑠)𝐶1𝑑𝜆(𝑠).(3.7) Consequently 𝑉 is bounded on 𝐿1(𝐷,𝑑𝜆,2). For 1<𝑝< we have that (𝑉𝐴)(𝑟)𝐶𝑝10𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘𝑠𝑘1𝑟221𝑠22𝐴𝑘(𝑠)𝐶𝑝=𝑑𝜆(𝑠)10(𝐴𝑠)𝑇(𝑟𝑠)𝐶𝑝𝑑𝜆(𝑠),(3.8) where 𝑇(𝑟𝑠)=(𝑡𝑗𝑖(𝑟𝑠))𝑖,𝑗 is a Toeplitz matrix and 𝑡𝑗𝑖((𝑟𝑠)=𝑟𝑠)𝑗𝑖1𝑠221𝑟22(𝑗𝑖+3)(𝑗𝑖+2)(𝑗𝑖+1)2if𝑗𝑖0otherwise.(3.9)𝑇(𝑟𝑠)is a Schur multiplier, therefore 10𝐴(𝑠)𝑇(𝑟𝑠)𝐶𝑝𝑑𝜆(𝑠)10𝐴(𝑠)𝐶𝑝1𝑟221𝑠2261𝑟𝑠𝑒𝑖𝜃4𝐿1(𝕋)=𝑆𝐴(𝑟).(3.10) From Schur’s theorem, (see, e.g., [8]) we obtain that 𝑆𝐴(𝑟) is bounded on 𝐿𝑝([0,1),𝑑𝜆). Hence 𝑉 is bounded on 𝐿𝑝(𝐷,𝑑𝜆,2), 1𝑝<, and there is a constant 𝐶>0 such that 𝑉(𝐴())𝐿𝑝(𝐷,𝑑𝜆,2)𝐶𝐵()𝐿𝑝(𝐷,𝑑𝜆,2)(3.11) for all 𝐴=𝑃𝐵(). Taking the infimum over 𝐵, we get that 𝑉(𝐴)𝐿𝑝(𝐷,𝑑𝜆,2)𝐶𝐴𝐵𝑝(2).(3.12) Thus 𝑉𝐵𝑝(2)𝐿𝑝(𝐷,𝑑𝜆,2) is bounded.
On the other hand, since 𝑃𝑉=𝑃 and 𝑉𝑃=𝑉 on 𝐿𝑝(𝐷,𝑑𝜆,2) we get easily that 𝐴=𝑃𝑉(𝐴) for all 𝐴𝐵𝑝(2). Thus 𝐴𝐵𝑝(2)=inf𝐵()𝐿𝑝(𝐷,𝑑𝜆,2)𝐴=𝑃𝐵𝑉𝐴𝐿𝑝(𝐷,𝑑𝜆,2),(3.13) and hence 𝑉𝐵𝑝(2)𝐿𝑝(𝐷,𝑑𝜆,2) is an embedding. The proof is complete.

We denote by 0,𝑐(𝐷,2) the closed Banach subspace of 0(𝐷,2) 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 𝐴,𝐵=10[]tr𝑉(𝐴)𝑉(𝐵)𝑑𝜆(𝑟)(3.14) One has the following dualities:(1)𝐵𝑝(2)𝐵𝑞(2) if 1<𝑝< and 1/𝑝+1/𝑞=1;(2)0,𝑐(𝐷,2)𝐵1(2) and 𝐵1(2)(𝐷,2).

Proof. Since 𝑉 is an embedding from 𝐵𝑝(2) into 𝐿𝑝(𝐷,𝑑𝜆,2) for all 1𝑝<, Hölder’s inequality shows that 𝐵𝑞(2)𝐵𝑝(2) for 1𝑝< and 𝐵1(2)0,𝑐(𝐷,2).
Suppose that 𝐹 is a bounded linear functional on the Besov-Schatten space 𝐵𝑝(2) with 1𝑝<. Then 𝐹𝑉1𝑉𝐵𝑝(2) extends to a bounded linear functional on 𝐿𝑝(𝐷,𝑑𝜆,2). Thus there exists 𝐶()𝐿𝑞(𝐷,𝑑𝜆,2) such that 𝐶()𝐿𝑞(𝐷,𝑑𝜆,2)=𝐹𝑉1 and 𝐹𝑉1(𝐵)=10[]tr(𝐵(𝑟))𝐶(𝑟)𝑑𝜆(𝑟),𝐵()𝐿𝑝𝐷,𝑑𝜆,2.(3.15) In particular, if 𝐵()=𝑉(𝐴) with 𝐴𝐵𝑝(2), then 𝐹(𝐴)=10[]tr((𝑉𝐴)(𝑟))𝐶(𝑟)𝑑𝜆(𝑟).(3.16) Let 𝐵=𝑃(𝐶). Then 𝐵𝐵𝑞(2) and it is easy to check that 𝐹(𝐴)=10[]tr(𝑉𝐴)(𝑟)(𝑉𝐵)(𝑟)𝑑𝜆(𝑟),𝐴𝐵𝑝2,(3.17) with 𝐵𝐵𝑞(2)𝐶()𝐿𝑝(𝐷,𝑑𝜆,2)=𝐹𝑉1𝑉1𝐹. This proves the duality 𝐵𝑝(2)𝐵𝑞(2) for 1𝑝<.
It remains to prove the duality 0,𝑐(𝐷,2)𝐵1(2).
Let us assume that 𝐹 is a bounded linear functional on 0,𝑐(𝐷,2). Then we will prove that there is a matrix 𝐶 from 𝐵1(2) such that 𝐹(𝐵)=10tr𝑉𝐵(𝑟)(𝑉𝐶)(𝑟)𝑑𝜆(𝑟),(3.18) for 𝐵 from a dense subset of 0(𝐷,2). By Lemma 1.2 it follows that 𝑉0(𝐷,2)𝒞0(𝐷,2) is an isomorphic embedding. Thus 𝑋=𝑉(0,𝑐(𝐷,2)) is a closed subspace in 𝒞0(𝐷,𝐶) and 𝐹(𝑉)1𝑋 is a bounded linear functional on 𝑋, where 0(𝐷,𝐶) is the subset in 𝒞0(𝐷,2) whose elements are 𝐶-valued functions. By the Hahn-Banach theorem 𝐹(𝑉)1 can be extended to a bounded linear functional on 𝒞0(𝐷,𝐶).
Let Φ𝒞0(𝐷,𝐶) denote this functional. It follows that 𝒞0(𝐷,𝐶)=𝒞0[0,1]𝜖𝐶 and, thus, Φ is a bilinear integral map, that is, there is a bounded Borel measure 𝜇 on [0,1]×𝑈𝐶1, where 𝑈𝐶1 is the unit ball of the space 𝐶1 with the topology 𝜎(𝐶1,𝐶), such that Φ(𝑓𝐴)=[0,1]×𝑈𝐶1𝑓(𝑟)tr𝐴𝐵𝑑𝜇(𝑟,𝐵)(3.19) for every 𝑓𝒞0[0,1] and 𝐴𝐶.
Thus, for the matrix 𝑛𝑘=0𝐴𝑘0,𝑐(𝐷,2), identified with the analytic matrix 𝑛𝑘=0𝐴𝑘𝑟𝑘, we have that 𝐹𝑛𝑘=0𝐴𝑘=𝐹𝑛𝑘=0𝑟𝑘𝐴𝑘=𝐹(𝑉)1𝑉𝑛𝑘=0𝑟𝑘𝐴𝑘=Φ𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑟𝑘1𝑟22𝐴𝑘=[0,1]×𝑈𝐶1𝑛𝑘=0tr(𝑘+3)(𝑘+2)2𝑟𝑘1𝑟22𝐴𝑘𝐵𝑑𝜇(𝑟,𝐵)def=𝜇(𝑟,𝐵),tr𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑟𝑘𝐴𝑘𝐵1𝑟22.(3.20) On the other hand, we wish to have that 𝐹(𝐴)=10tr𝑉(𝐴)(𝑉(𝐶))=𝑑𝜆(𝑠)10tr𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑠𝑘𝐴𝑘(𝑉(𝐶))=(2𝑠𝑑𝑠)10tr𝑛𝑘=0𝑠2𝑘(𝑘+3)2(𝑘+2)241𝑠22𝐴𝑘𝐶𝑘=(2𝑠𝑑𝑠)𝑛𝑘=0tr𝐴𝑘(𝑘+3)(𝑘+2)𝐶2(𝑘+1)𝑘.(3.21) Therefore, letting 𝐴=𝑒𝑖,𝑖+𝑘, denote the matrix having 1 as the single nonzero entry on the 𝑖th-row and the (𝑖+𝑘)th-column, for 𝑖1 and 𝑗0, we have that 𝐶𝑘=𝜇(𝑟,𝐵),(𝑘+1)𝑟𝑘1𝑟22𝐵𝑘,𝑘=0,1,2,.(3.22) Then, it yields that 10𝐶(𝑠)𝐶1=2𝑠𝑑𝑠10[0,1]×𝑈𝐶1𝑛𝑘=2(𝑘+1)!𝑠(𝑘2)!𝑘2𝑟𝑘1𝑟22𝐵𝑘𝑑𝜇(𝑟,𝐵)𝐶1(2𝑠𝑑𝑠)[0,1]×𝑈𝐶110𝑛𝑘=2(𝑘+1)!(𝑘2)!(𝑟𝑠)𝑘2𝑟21𝑟22𝐵𝑘𝐶1𝑑||𝜇||(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶110𝑛𝑘=2(𝑘+1)!(𝑘2)!(𝑟𝑠)𝑘2𝑟21𝑟22𝑒𝑖𝑘()𝐿1(𝕋)𝐵𝐶1𝑑||𝜇||(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶11002𝜋𝑟21𝑟22||1𝑟𝑠𝑒𝑖𝜃||4𝑑𝜃𝑑||𝜇||2𝜋(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶1𝑟21𝑟2211𝑟22𝑑||𝜇||(𝑟,𝐵)𝜇<.(3.23) Consequently, 𝐶𝐵1(2) and we get the relation (3.18) by using the fact that the set of all matrices 𝑛𝑘=0𝐴𝑘 is dense in 0,𝑐(𝐷,2).

As an application of the description of the dual space of Besov-Schatten space we give a characterization of the space of all Schur multipliers between Besov-Schatten spaces 𝐵1(2).

Theorem 3.3. One has(𝐵1(2),𝐵1(2))=𝐻11,,1(2)def={𝐴sup𝑟<1(1𝑟)𝑘0𝑘𝐴𝑘𝑀(2)<}.

Proof. By Lemma 3.1 we have that 𝑉(𝐴𝐵)=𝑉(𝐴)𝐵 for all 𝐴𝐵1(2) and for all matrices 𝐵 such that 𝐴𝐵𝐵1(2). Consequently (𝐵1(2),𝐵1(2))=((2),(2)). Finally, by using [12, Theorem  6] we get the stated result.

Acknowledgments

The authors want to thank Professor Nicolae Popa for his helpful suggestions that have contributed to improve the final version of this paper. A. N. Marcoci and L. G. Marcoci were partially supported by CNCSIS-UEFISCSU, project number 538/2009 PNII-IDEI code 1905/2008.

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Copyright © 2012 A. N. Marcoci et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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