We introduce the Besov-Schatten spaces 𝐵𝑝(ℓ2), 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(ℓ2).

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., [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 𝐴=(ğ‘Žğ‘—ğ‘˜)𝑗,𝑘≥0 and 𝐵=(𝑏𝑗𝑘)𝑗,𝑘≥0 is defined as the matrix 𝐴∗𝐵 whose entries are the products of the entries of 𝐴 and 𝐵: î€·ğ‘Žğ´âˆ—ğµ=𝑗𝑘𝑏𝑗𝑘𝑗,𝑘≥0.(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)={𝑀∶𝑀∗𝐴∈𝑌forevery𝐴∈𝑋},(1.2) equipped with the natural norm ‖𝑀‖=sup‖𝐴‖𝑋≤1‖𝑀∗𝐴‖𝑌.(1.3) In the case 𝑋=𝑌=𝐵(ℓ2), where 𝐵(ℓ2) is the space of all linear and bounded operators on ℓ2, the space 𝑀(𝐵(ℓ2),𝐵(ℓ2)) will be denoted 𝑀(ℓ2) and a matrix 𝐴∈𝑀(ℓ2) 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=(𝑐𝑗−𝑘)𝑗,𝑘, where (𝑐𝑛)𝑛∈ℤ 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(ğ¶ğ‘î…ž), where 1/𝑝+1/𝑝′=1 and 𝑀(ℓ2)=𝑀(𝐶1). Next, interpolating between the classes 𝐶𝑝, we can easily see that 𝑀(𝐶𝑝1)⊂𝑀(𝐶𝑝2) if 0<𝑝1≤𝑝2≤2 (see, e.g., [9]). We will denote by 𝐴𝑘, the 𝑘th-diagonal matrix associated to 𝐴 (see [4]). For an infinite matrix 𝐴=(ğ‘Žğ‘–ğ‘—) and an integer 𝑘 we denote by 𝐴𝑘 the matrix whose entries ğ‘Žâ€²ğ‘–ğ‘— are given by ğ‘Žâ€²ğ‘–ğ‘—=î‚»ğ‘Žğ‘–ğ‘—if𝑗−𝑖=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)=âˆžî“ğ‘˜=âˆ’âˆžğ´ğ‘˜(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∈ℤ. 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 ℬ(𝐷,ℓ2) is the space of all analytic matrices 𝐴 with 𝐴(𝑟)∈𝐵(ℓ2), 0≤𝑟<1, such that ‖𝐴‖ℬ(𝐷,ℓ2)=sup0≤𝑟<11−𝑟2î€¸â€–â€–ğ´î…žâ€–â€–(𝑟)𝐵(ℓ2)+‖‖𝐴0‖‖𝐵(ℓ2)<∞,(1.8) where 𝐵(ℓ2) is the usual operator norm of the matrix 𝐴 on the sequence space ℓ2 and∑𝐴′(𝑟)=âˆžğ‘˜=0𝐴𝑘𝑘𝑟𝑘−1.

The space ℬ0(𝐷,ℓ2) is the space of all upper triangular infinite matrices 𝐴 such that lim𝑟→1−(1−𝑟2)‖(𝐴∗𝐶(𝑟))′‖𝐵(ℓ2)=0, where 𝐶(𝑟) is the Toeplitz matrix associated with the Cauchy kernel 1/(1−𝑟), for 0≤𝑟<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 𝑟→𝐴(𝑟) defined on [0,1) with ∫10‖𝐴(𝑟)‖𝑝𝐶𝑝2𝑟𝑑𝑟<∞ and for all 𝑖,𝑗∈ℕ we have that []âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ƒ(𝐴(⋅))(𝑟)(𝑖,𝑗)=2(𝑗−𝑖+1)𝑟𝑗−𝑖10ğ‘Žğ‘–ğ‘—(𝑠)⋅𝑠𝑗−𝑖+1𝑑𝑠,if𝑖≤𝑗,0,otherwise.(1.9)

Now we consider a modified version of Bergman projection.

Let 𝛼>−1. Thenî€ºğ‘ƒğ›¼î€»âŽ§âŽªâŽ¨âŽªâŽ©ğ´(⋅)(𝑟)=(𝛼+1)Γ(𝑗−𝑖+2+𝛼)𝑟(𝑗−𝑖)!Γ(𝛼+2)𝑗−𝑖210ğ‘Žğ‘–ğ‘—(𝑠)𝑠𝑗−𝑖+11−𝑠2𝛼𝑑𝑠if𝑗≥𝑖0if𝑗<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)(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)2𝑟𝑗−𝑖1−𝑟2210ğ‘Žğ‘–ğ‘—(𝑠)𝑠𝑗−𝑖(2𝑠𝑑𝑠)if𝑗−𝑖≥00otherwise.(1.11)
Then 𝑉 is an isomorphic embedding of ℬ0(𝐷,ℓ2) in ğ’ž0(𝐷,ℓ2), where ğ’ž0(𝐷,ℓ2) is the space of all continuous 𝐵(ℓ2)-valued functions 𝐵(𝑟) on [0,1) such that lim𝑟→1𝐵(𝑟)=0 in the norm of 𝐵(ℓ2).

The paper is organized as follows. In Section 2 we give a characterization of matrices in the Besov-Schatten space 𝐵𝑝(ℓ2) 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 1≤𝑝<∞ and a positive measure on [0,1) given by 𝑑𝜆(𝑟)=2𝑟𝑑𝑟1−𝑟22.(2.1) The Besov-Schatten matrix space 𝐵𝑝(ℓ2) is defined to be the space of all upper triangular infinite matrices 𝐴 such that ‖𝐴‖𝐵𝑝(ℓ2)=101−𝑟22ğ‘â€–â€–ğ´î…žî…žâ€–â€–(𝑟)𝑝𝐶𝑝𝑑𝜆(𝑟)1/𝑝<∞.(2.2)

On 𝐵𝑝(ℓ2) we introduce the norm ‖‖𝐴‖𝐴‖=0‖‖𝐶1+‖𝐴‖𝐵𝑝(ℓ2).(2.3)

We introduce the notation 𝐿𝑝(𝐷,𝑑𝜆,ℓ2) for the space of all strongly measurable functions 𝑟→𝐴(𝑟) defined on the measurable space ([0,1),𝑑𝜆) with 𝐶𝑝 values such that ‖𝐴‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2)=10‖𝐴(𝑟)‖𝑝𝐶𝑝𝑑𝜆(𝑟)1/𝑝<∞.(2.4)

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

Lemma 2.2. Let 𝑧∈𝐷, 𝑐 is real, 𝑡>−1, and 𝐼𝑐,𝑡=𝐷1−|𝑤|2𝑡||1−𝑧𝑤||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|2𝑐(|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 1≤𝑝<∞ and 𝐴 be an upper triangular matrix such that the 𝐶𝑝-valued function ğ‘Ÿâ†’ğ´î…žî…ž(𝑟) is continuous on [0,𝑟0) for some 1>𝑟0>0. Then the following assertions are equivalent:(1)𝐴∈𝐵𝑝(ℓ2);(2)(1−𝑟2)2ğ´î…žî…ž(𝑟)∈𝐿𝑝(𝐷,𝑑𝜆,ℓ2);(3)𝐴∈𝑃𝐿𝑝(𝐷,𝑑𝜆,ℓ2), 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(⋅))=âˆžî“ğ‘˜=0(𝑘+1)10[]𝐴(𝑠)𝑘𝑠𝑘(2𝑠𝑑𝑠),(2.8) where 𝐴(⋅)∈𝐿𝑝(𝐷,ℓ2). Then 𝑃1−𝑟22𝐴𝑘𝑟𝑘(𝑠)=2𝑠𝑘𝐴𝑘,(𝑘+2)(𝑘+3)(2.9) for all 𝑘≥0, and all 𝐴𝑘∈𝐶𝑝.
It follows that each matriceal polynomial is in 𝑃𝐿𝑝(𝐷,𝑑𝜆,ℓ2) for all 1≤𝑝<∞.
Suppose that 𝐴 is an upper triangular matrix with 𝐴𝑘∈𝐶𝑝 for all 𝑘≥0. We write 𝐴=4𝑘=0𝐴𝑘+𝐴1,(2.10) where 𝐴1∑∶=âˆžğ‘˜=5𝐴𝑘.
If (1−𝑟2)2ğ´î…žî…ž(𝑟)∈𝐿𝑝(𝐷,𝑑𝜆,ℓ2), then we have that Φ(𝑟)∶=4𝑘=0(𝑘+2)(𝑘+3)21−𝑟22𝐴𝑘𝑟𝑘+1−𝑟22𝐴1(𝑟)2!𝑟2(2.11) is in 𝐿𝑝(𝐷,𝑑𝜆,ℓ2) and moreover that 𝐴=𝑃Φ.
Indeed, for 0<𝑟<𝑟0, 𝑟→(𝐴1)(𝑟) is a continuous function and, therefore 𝑟00‖‖𝐴1‖‖(𝑠)𝑝𝐶𝑝𝑠2𝑝𝑑𝑠<∞.(2.12) Consequently Φ∈𝐿𝑝(𝐷,𝑑𝜆,ℓ2).
Moreover 𝐴=𝑃Φ since âˆžî“ğ‘˜=510𝑘(𝑘−1)𝐴𝑘𝑠𝑘−21−𝑠22𝑠2(𝑘+1)𝑠𝑘+1=ğ‘‘ğ‘ âˆžî“ğ‘˜=5(𝑘−1)𝑘(𝑘+1)𝐴𝑘10𝑠2𝑘−31−𝑠22𝑑𝑠=âˆžî“ğ‘˜=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,ℓ2). Then we have that 1−𝑟22ğ´î…žî…žî€·(𝑟)=1−𝑟2210Φ(𝑠)∗6𝑠2(1−𝑟𝑠)42𝑠𝑑𝑠.(2.14) Using Fubini’s theorem and Lemma 2.2 we obtain that 101−𝑟22â€–â€–ğ´î…žî…žâ€–â€–(𝑟)𝐶12𝑟𝑑𝑟1−𝑟22≤1010‖Φ(𝑠)‖𝐶102𝜋6𝑠2𝑑𝜃||1−𝑟𝑠𝑒𝑖𝜃||4=2𝑠𝑑𝑠2𝑟𝑑𝑟106𝑠2(‖Φ𝑠)‖𝐶11012𝜋02𝜋𝑑𝜃||1−𝑟𝑠𝑒𝑖𝜃||4∼2𝑟𝑑𝑟2𝑠𝑑𝑠10‖Φ(𝑠)‖𝐶112𝑠31−𝑠22𝑑𝑠≤610‖Φ(𝑠)‖𝐶1𝑑𝜆(𝑠)<∞.(2.15)
Consequently, 𝐴∈𝐿1(𝐷,𝑑𝜆,ℓ2) 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))âˆžğ‘–,𝑗=1 be the Toeplitz matrix with 𝑡𝑖𝑗(𝑟𝑠)=𝑡𝑗−𝑖𝑠(𝑟𝑠)=2(𝑟𝑠)𝑗−𝑖(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)if𝑗≥𝑖0otherwise.(2.16) Since 𝑇(𝑟𝑠) is a Schur multiplier with ‖𝑇(𝑟𝑠)‖𝑀(ℓ2)=‖𝑇(𝑟𝑠)‖𝐿1(𝕋)=‖6𝑠2/(1−𝑟𝑠𝑒𝑖𝜃)4‖𝐿1(𝕋) and 𝑀(ℓ2)=𝑀(𝐶1)⊂𝑀(𝐶𝑝), 1≤𝑝<∞ we get that 1−𝑟22â€–â€–ğ´î…žî…žâ€–â€–(𝑟)𝐶𝑝=1−𝑟22‖‖‖10𝜙(𝑠)∗6𝑠2(1−𝑟𝑠)4‖‖‖(2𝑠𝑑𝑠)𝐶𝑝≤1−𝑟2210‖‖𝜙(𝑠)𝐶𝑝‖‖‖‖6𝑠21−𝑟𝑠𝑒𝑖𝜃4‖‖‖‖𝐿1(𝕋)=(2𝑠𝑑𝑠)1−𝑟2210‖‖𝜙(𝑠)𝐶𝑝1−𝑠22‖‖‖‖6𝑠21−𝑟𝑠𝑒𝑖𝜃4‖‖‖‖𝐿1(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 1−𝑟22ğ´î…žî…ž(𝑟)∈𝐿𝑝𝐷,𝑑𝜆,ℓ2(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, []=âŽ§âŽªâŽ¨âŽªâŽ©ğ‘‰(𝐴(⋅))(𝑟)(𝑖,𝑗)(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)2𝑟𝑗−𝑖1−𝑟2210ğ‘Žğ‘–ğ‘—(𝑠)𝑠𝑗−𝑖(2𝑠𝑑𝑠)if𝑗−𝑖≥0,0otherwise.(3.1)
Then 𝑉 is an embedding from 𝐵𝑝(ℓ2) into 𝐿𝑝(𝐷,𝑑𝜆,ℓ2) for all 𝑝≥1, if 𝐵𝑝(ℓ2)=𝑃𝐿𝑝(𝐷,𝑑𝜆,ℓ2) is equipped with the quotient norm.

Proof. Suppose that 𝐴∈𝐵𝑝(ℓ2) and 𝐵(⋅)∈𝐿𝑝(𝐷,𝑑𝜆,ℓ2) with 𝐴=𝑃𝐵(⋅). Since âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ƒ(𝐵(⋅))(𝑟)(𝑖,𝑗)=2(𝑗−𝑖+1)𝑟𝑗−𝑖10𝑏𝑖𝑗(𝑠)𝑠𝑗−𝑖+1𝑑𝑠if𝑗−𝑖≥00otherwise,(3.2) it is easy to see that 𝑃𝑉=𝑃,𝑉𝑃=𝑉(3.3) on 𝐿𝑝(𝐷,𝑑𝜆,ℓ2). Therefore 𝑉(𝐴)=𝑉(𝐵(⋅)) for all 𝐴∈𝐵𝑝(ℓ2) and 𝐵(⋅)∈𝐿𝑝(𝐷,𝑑𝜆,ℓ2).
We will now prove that 𝑉 is a bounded operator on𝐿𝑝(𝐷,𝑑𝜆,ℓ2). We first prove this fact for 𝑝=1. By Fubini’s theorem we have that ‖‖𝑉(𝐴(⋅))𝐿1(𝐷,𝑑𝜆,ℓ2)=10‖[]‖𝑉(𝐴(⋅))𝐶1=𝑑𝜆(𝑟)10â€–â€–â€–â€–âˆžî“ğ‘˜=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘1−𝑟2210𝐴𝑘(𝑠)𝑠𝑘(‖‖‖‖2𝑠𝑑𝑠)𝐶1≤𝑑𝜆(𝑟)10â€–â€–â€–â€–âˆžî“ğ‘˜=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘(1−𝑟2)2𝐴𝑘(𝑠)𝑠𝑘+1‖‖‖‖𝐶1=(2𝑑𝑠)𝑑𝜆(𝑟)10⎡⎢⎢⎣10â€–â€–â€–â€–âˆžî“ğ‘˜=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘𝐴𝑘(𝑠)𝑠𝑘1−𝑠22‖‖‖‖𝐶1⎤⎥⎥⎦=(2𝑟𝑑𝑟)𝑑𝜆(𝑠)10‖𝐴(𝑠)∗𝐶(𝑟𝑠)‖𝐶1(2𝑟𝑑𝑟)𝑑𝜆(𝑠),(3.4) where 𝐶(𝑟𝑠)=(𝑐𝑖𝑗(𝑟𝑠))âˆžğ‘–,𝑙=1 means the Toeplitz matrix given by 𝑐𝑖𝑗(𝑟𝑠)=𝑐𝑗−𝑖=((𝑟𝑠)𝑟𝑠)𝑗−𝑖𝑠21−𝑠22(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)2if𝑗≥𝑖,0otherwise.(3.5) Since the Toeplitz matrix 𝐶(𝑟𝑠) is a Schur multiplier with ‖𝐶(𝑟𝑠)‖𝑀(ℓ2)=‖‖‖‖6𝑠21−𝑠221−𝑟𝑠𝑒𝑖𝜃4‖‖‖‖𝐿1(𝕋),(3.6) then, according to Lemma 2.2, it follows that 10‖‖𝐴(𝑠)∗𝐶(𝑟𝑠)𝐶1(2𝑟𝑑𝑟)𝑑𝜆(𝑠)≤10‖‖𝐴(𝑠)𝐶110‖‖𝐶(𝑟𝑠)𝑀(ℓ2)∼(2𝑟𝑑𝑟)𝑑𝜆(𝑠)10‖‖𝐴(𝑠)𝐶1𝑑𝜆(𝑠).(3.7) Consequently 𝑉 is bounded on 𝐿1(𝐷,𝑑𝜆,ℓ2). For 1<𝑝<∞ we have that (‖𝑉𝐴⋅)(𝑟)‖𝐶𝑝≤10â€–â€–â€–â€–âˆžî“ğ‘˜=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟𝑘𝑠𝑘1−𝑟221−𝑠22𝐴𝑘(‖‖‖‖𝑠)𝐶𝑝=𝑑𝜆(𝑠)10(‖𝐴𝑠)∗𝑇(𝑟𝑠)‖𝐶𝑝𝑑𝜆(𝑠),(3.8) where 𝑇(𝑟𝑠)=(𝑡𝑗−𝑖(𝑟𝑠))𝑖,𝑗 is a Toeplitz matrix and 𝑡𝑗−𝑖((𝑟𝑠)=𝑟𝑠)𝑗−𝑖1−𝑠221−𝑟22(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)2if𝑗≥𝑖0otherwise.(3.9)𝑇(𝑟𝑠)is a Schur multiplier, therefore 10‖𝐴(𝑠)∗𝑇(𝑟𝑠)‖𝐶𝑝𝑑𝜆(𝑠)≤10‖𝐴(𝑠)‖𝐶𝑝1−𝑟221−𝑠22‖‖‖‖61−𝑟𝑠𝑒𝑖𝜃4‖‖‖‖𝐿1(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,ℓ2), 1≤𝑝<∞, and there is a constant 𝐶>0 such that ‖𝑉(𝐴(⋅))‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2)≤𝐶‖𝐵(⋅)‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2)(3.11) for all 𝐴=𝑃𝐵(⋅). Taking the infimum over 𝐵, we get that ‖𝑉(𝐴)‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2)≤𝐶‖𝐴‖𝐵𝑝(ℓ2).(3.12) Thus 𝑉∶𝐵𝑝(ℓ2)→𝐿𝑝(𝐷,𝑑𝜆,ℓ2) is bounded.
On the other hand, since 𝑃𝑉=𝑃 and 𝑉𝑃=𝑉 on 𝐿𝑝(𝐷,𝑑𝜆,ℓ2) we get easily that 𝐴=𝑃𝑉(𝐴) for all 𝐴∈𝐵𝑝(ℓ2). Thus ‖𝐴‖𝐵𝑝(ℓ2)‖=inf‖𝐵(⋅)𝐿𝑝(𝐷,𝑑𝜆,ℓ2)∶𝐴=𝑃𝐵≤‖𝑉𝐴‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2),(3.13) and hence 𝑉∶𝐵𝑝(ℓ2)→𝐿𝑝(𝐷,𝑑𝜆,ℓ2) is an embedding. The proof is complete.

We denote by ℬ0,𝑐(𝐷,ℓ2) the closed Banach subspace of ℬ0(𝐷,ℓ2) 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(ℓ2)âˆ—â‰ˆğµğ‘ž(ℓ2) if 1<𝑝<∞ and 1/𝑝+1/ğ‘ž=1;(2)ℬ0,𝑐(𝐷,ℓ2)∗≈𝐵1(ℓ2) and 𝐵1(ℓ2)∗≈ℬ(𝐷,ℓ2).

Proof. Since 𝑉 is an embedding from 𝐵𝑝(ℓ2) into 𝐿𝑝(𝐷,𝑑𝜆,ℓ2) for all 1≤𝑝<∞, Hölder’s inequality shows that ğµğ‘ž(ℓ2)⊂𝐵𝑝(ℓ2)∗ for 1≤𝑝<∞ and 𝐵1(ℓ2)⊂ℬ∗0,𝑐(𝐷,ℓ2).
Suppose that 𝐹 is a bounded linear functional on the Besov-Schatten space 𝐵𝑝(ℓ2) with 1≤𝑝<∞. Then 𝐹∘𝑉−1∶𝑉𝐵𝑝(ℓ2)→ℂ extends to a bounded linear functional on 𝐿𝑝(𝐷,𝑑𝜆,ℓ2). Thus there exists 𝐶(⋅)âˆˆğ¿ğ‘ž(𝐷,𝑑𝜆,ℓ2) such that ‖𝐶(⋅)â€–ğ¿ğ‘ž(𝐷,𝑑𝜆,ℓ2)=‖𝐹∘𝑉−1‖ and 𝐹∘𝑉−1(𝐵)=10[]tr(𝐵(𝑟))𝐶(𝑟)∗𝑑𝜆(𝑟),𝐵(⋅)∈𝐿𝑝𝐷,𝑑𝜆,ℓ2.(3.15) In particular, if 𝐵(⋅)=𝑉(𝐴) with 𝐴∈𝐵𝑝(ℓ2), then 𝐹(𝐴)=10[]tr((𝑉𝐴)(𝑟))𝐶(𝑟)∗𝑑𝜆(𝑟).(3.16) Let 𝐵=𝑃(𝐶). Then ğµâˆˆğµğ‘ž(ℓ2) and it is easy to check that 𝐹(𝐴)=10[]tr(𝑉𝐴)(𝑟)(𝑉𝐵)(𝑟)∗𝑑𝜆(𝑟),𝐴∈𝐵𝑝ℓ2,(3.17) with â€–ğµâ€–ğµğ‘ž(ℓ2)≤‖𝐶(⋅)‖𝐿𝑝(𝐷,𝑑𝜆,ℓ2)=‖𝐹∘𝑉−1‖≤‖𝑉−1‖‖𝐹‖. This proves the duality 𝐵𝑝(ℓ2)âˆ—â‰ˆğµğ‘ž(ℓ2) for 1≤𝑝<∞.
It remains to prove the duality ℬ∗0,𝑐(𝐷,ℓ2)≈𝐵1(ℓ2).
Let us assume that 𝐹 is a bounded linear functional on ℬ0,𝑐(𝐷,ℓ2). Then we will prove that there is a matrix 𝐶 from 𝐵1(ℓ2) such that 𝐹(𝐵)=10tr𝑉𝐵(𝑟)(𝑉𝐶)∗(𝑟)𝑑𝜆(𝑟),(3.18) for 𝐵 from a dense subset of ℬ0(𝐷,ℓ2). By Lemma 1.2 it follows that 𝑉∶ℬ0(𝐷,ℓ2)â†’ğ’ž0(𝐷,ℓ2) is an isomorphic embedding. Thus 𝑋=𝑉(ℬ0,𝑐(𝐷,ℓ2)) is a closed subspace in ğ’ž0(𝐷,ğ¶âˆž) and 𝐹∘(𝑉)−1∶𝑋→ℂ is a bounded linear functional on 𝑋, where ℂ0(𝐷,ğ¶âˆž) is the subset in ğ’ž0(𝐷,ℓ2) whose elements are ğ¶âˆž-valued functions. By the Hahn-Banach theorem 𝐹∘(𝑉)−1 can be extended to a bounded linear functional on ğ’ž0(𝐷,ğ¶âˆž).
Let Î¦âˆ¶ğ’ž0(𝐷,ğ¶âˆž)→ℂ denote this functional. It follows that ğ’ž0(𝐷,ğ¶âˆž)=ğ’ž0⊗[0,1]ğœ–ğ¶âˆž 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 ğœŽ(𝐶1,ğ¶âˆž), such that Φ(𝑓⊗𝐴)=[0,1]×𝑈𝐶1𝑓(𝑟)tr𝐴𝐵∗𝑑𝜇(𝑟,𝐵)(3.19) for every ğ‘“âˆˆğ’ž0[0,1] and ğ´âˆˆğ¶âˆž.
Thus, for the matrix ∑𝑛𝑘=0𝐴𝑘∈ℬ0,𝑐(𝐷,ℓ2), identified with the analytic matrix ∑𝑛𝑘=0𝐴𝑘𝑟𝑘, we have that 𝐹𝑛𝑘=0𝐴𝑘=𝐹𝑛𝑘=0𝑟𝑘𝐴𝑘=𝐹∘(𝑉)−1𝑉𝑛𝑘=0𝑟𝑘𝐴𝑘=Φ𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑟𝑘1−𝑟22𝐴𝑘=[0,1]×𝑈𝐶1𝑛𝑘=0tr(𝑘+3)(𝑘+2)2𝑟𝑘1−𝑟22𝐴𝑘𝐵∗𝑑𝜇(𝑟,𝐵)def=𝜇(𝑟,𝐵),tr𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑟𝑘𝐴𝑘𝐵∗1−𝑟22.(3.20) On the other hand, we wish to have that 𝐹(𝐴)=10tr𝑉(𝐴)(𝑉(𝐶))∗=𝑑𝜆(𝑠)10tr𝑛𝑘=0(𝑘+3)(𝑘+2)2𝑠𝑘𝐴𝑘(𝑉(𝐶))∗=(2𝑠𝑑𝑠)10tr𝑛𝑘=0𝑠2𝑘(𝑘+3)2(𝑘+2)241−𝑠22𝐴𝑘𝐶∗𝑘=(2𝑠𝑑𝑠)𝑛𝑘=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 𝑖≥1 and 𝑗≥0, we have that 𝐶𝑘=𝜇(𝑟,𝐵),(𝑘+1)𝑟𝑘1−𝑟22𝐵𝑘,𝑘=0,1,2,….(3.22) Then, it yields that 10â€–â€–ğ¶î…žî…žâ€–â€–(𝑠)𝐶1=2𝑠𝑑𝑠10‖‖‖‖[0,1]×𝑈𝐶1𝑛𝑘=2(𝑘+1)!𝑠(𝑘−2)!𝑘−2𝑟𝑘1−𝑟22𝐵𝑘‖‖‖‖𝑑𝜇(𝑟,𝐵)𝐶1≤(2𝑠𝑑𝑠)[0,1]×𝑈𝐶1⎡⎢⎢⎣10‖‖‖‖𝑛𝑘=2(𝑘+1)!(𝑘−2)!(𝑟𝑠)𝑘−2𝑟21−𝑟22𝐵𝑘‖‖‖‖𝐶1âŽ¤âŽ¥âŽ¥âŽ¦ğ‘‘||𝜇||≤(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶1⎡⎢⎢⎣10‖‖‖‖𝑛𝑘=2(𝑘+1)!(𝑘−2)!(𝑟𝑠)𝑘−2𝑟21−𝑟22𝑒𝑖𝑘(⋅)‖‖‖‖𝐿1(𝕋)‖𝐵‖𝐶1âŽ¤âŽ¥âŽ¥âŽ¦ğ‘‘||𝜇||≤(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶11002𝜋𝑟21−𝑟22||1−𝑟𝑠𝑒𝑖𝜃||4𝑑𝜃𝑑||𝜇||∼2𝜋(2𝑠𝑑𝑠)(𝑟,𝐵)[0,1]×𝑈𝐶1𝑟21−𝑟2211−𝑟22𝑑||𝜇||(𝑟,𝐵)≤‖𝜇‖<∞.(3.23) Consequently, 𝐶∈𝐵1(ℓ2) and we get the relation (3.18) by using the fact that the set of all matrices ∑𝑛𝑘=0𝐴𝑘 is dense in ℬ0,𝑐(𝐷,ℓ2).

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(ℓ2).

Theorem 3.3. One has(𝐵1(ℓ2),𝐵1(ℓ2))=𝐻11,∞,1(ℓ2)def={𝐴∶sup𝑟<1∑(1−𝑟)‖𝑘≥0𝑘𝐴𝑘‖𝑀(ℓ2)<∞}.

Proof. By Lemma 3.1 we have that 𝑉(𝐴∗𝐵)=𝑉(𝐴)∗𝐵 for all 𝐴∈𝐵1(ℓ2) and for all matrices 𝐵 such that 𝐴∗𝐵∈𝐵1(ℓ2). Consequently (𝐵1(ℓ2),𝐵1(ℓ2))=(ℬ(ℓ2),ℬ(ℓ2)). Finally, by using [12, Theorem  6] we get the stated result.


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.