Besov-Schatten Spaces

Abstract

We introduce the Besov-Schatten spaces 𝐵_𝑝(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=(𝑎_𝑗𝑘)_{𝑗,𝑘≥0} and 𝐵=(𝑏_𝑗𝑘)_{𝑗,𝑘≥0} is defined as the matrix 𝐴∗𝐵 whose entries are the products of the entries of 𝐴 and 𝐵: 𝐴∗𝐵=𝑎_𝑗𝑘𝑏_𝑗𝑘_{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=(𝑐_𝑗−𝑘)_𝑗,𝑘, where (𝑐_𝑛)_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)=𝑐_𝑛,for𝑛=0,±1,±2,….(1.4) Moreover, one then has that ‖𝑀‖=‖𝜇‖.(1.5)

We will denote by 𝐶_𝑝, 0<𝑝<∞, the Schatten class operators (see, e.g., [8]). Let us summarize briefly some well-known properties of classes 𝑀(𝐶_𝑝) which will be very often used in what follows. If 1<𝑝<∞, then 𝑀(𝐶_𝑝)=𝑀(𝐶_𝑝), where 1/𝑝+1/𝑝′=1 and 𝑀(ℓ²)=𝑀(𝐶₁). Next, interpolating between the classes 𝐶_𝑝, we can easily see that 𝑀(𝐶_𝑝1)⊂𝑀(𝐶_𝑝2) if 0<𝑝₁≤𝑝₂≤2 (see, e.g., [9]). We will denote by 𝐴_𝑘, the 𝑘th-diagonal matrix associated to 𝐴 (see [4]). For an infinite matrix 𝐴=(𝑎_𝑖𝑗) and an integer 𝑘 we denote by 𝐴_𝑘 the matrix whose entries 𝑎′_𝑖𝑗 are given by 𝑎′_𝑖𝑗=𝑎^𝑖𝑗0oifth𝑗e−rw𝑖i=se.𝑘,(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)⟶𝑓_𝐴(𝑟,𝑡)=^∞_𝑘_=−∞𝐴_𝑘(𝑟)𝑒^𝑖𝑘𝑡,(1.7) where 𝐴_𝑘(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 𝐴_𝑘(𝑟)=𝐴_𝑘𝑟^𝑘, 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₎=sup_0≤𝑟<11−𝑟²‖‖𝐴^(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 and𝐴′(𝑟)=∑^∞𝑘=0𝐴_𝑘𝑘𝑟^𝑘−1.

The space ℬ₀(𝐷,ℓ²) is the space of all upper triangular infinite matrices 𝐴 such that lim_𝑟→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 𝐶_𝑝-valued functions 𝑟→𝐴(𝑟) defined on [0,1) with ∫¹⁰‖𝐴(𝑟)‖^𝑝𝐶_𝑝2𝑟𝑑𝑟<∞ and for all 𝑖,𝑗∈ℕ we have that [𝑃(𝐴(⋅))](𝑟)(𝑖,𝑗)=⎧⎪⎨⎪⎩2(𝑗−𝑖+1)𝑟^𝑗−𝑖¹⁰0,𝑎_𝑖𝑗(𝑠)⋅𝑠^{𝑗−𝑖+1}𝑑𝑠,iofth𝑖e≤rw𝑗i,se.(1.9)

Now we consider a modified version of Bergman projection.

Let 𝛼>−1. Then𝑃_𝛼𝐴(⋅)(𝑟)=⎧⎪⎨⎪⎩(𝛼+(𝑗1−)Γ𝑖)(!𝑗Γ−(𝑖𝛼++22+)𝛼)𝑟^𝑗−𝑖2¹⁰0𝑎_𝑖𝑗(𝑠)𝑠^{𝑗−𝑖+1}1−𝑠²^𝛼𝑑𝑠iiff𝑗𝑗≥<𝑖𝑖.(1.10)

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

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

Lemma 1.2. Let 𝑉=(𝑃₂)^∗, that is, 𝑃₂𝐴=(⋅⎧⎪⎨⎪⎩)(^∗𝑗(−𝑟)𝑖(𝑖+,𝑗3))(𝑗−𝑖+2)(𝑗−𝑖+1)2𝑟^𝑗−𝑖1−𝑟²²¹⁰0𝑎_𝑖𝑗(𝑠)𝑠^𝑗−𝑖(2𝑠𝑑𝑠)iofth𝑗e−rw𝑖i≥se.0(1.11)
Then 𝑉 is an isomorphic embedding of ℬ₀(𝐷,ℓ²) in 𝒞₀(𝐷,ℓ²), where 𝒞₀(𝐷,ℓ²) is the space of all continuous 𝐵(ℓ²)-valued functions 𝐵(𝑟) on [0,1) such that lim_𝑟→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 𝐵_𝑝(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 𝐵_𝑝(ℓ²) is defined to be the space of all upper triangular infinite matrices 𝐴 such that ‖𝐴‖_𝐵𝑝(ℓ²₎=¹⁰1−𝑟²^2𝑝‖‖𝐴^(𝑟)‖‖^𝑝𝐶_𝑝𝑑𝜆(𝑟)^1/𝑝<∞.(2.2)

On 𝐵_𝑝(ℓ²) we introduce the norm ‖𝐴‖=‖‖𝐴₀‖‖_𝐶1+‖𝐴‖_𝐵𝑝(ℓ²₎.(2.3)

We introduce the notation 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) for the space of all strongly measurable functions 𝑟→𝐴(𝑟) defined on the measurable space ([0,1),𝑑𝜆) with 𝐶_𝑝 values such that ‖𝐴‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎=¹⁰‖𝐴(𝑟)‖^𝑝𝐶_𝑝𝑑𝜆(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 𝐼_𝑐,𝑡=_𝐷1−|𝑤|²^𝑡||1−𝑧𝑤||^2+𝑡+𝑐𝑑𝐴(𝑤).(2.5) Then,(1)if 𝑐<0, then 𝐼_𝑐,𝑡(𝑧) is bounded in 𝑧;(2)if 𝑐>0, then 𝐼_𝑐,𝑡(𝑧)∼11−|𝑧|²^𝑐(|𝑧|⟶1⁻);(2.6)(3)if 𝑐=0, then 𝐼_0,𝑐(𝑧)∼log11−|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 𝐶_𝑝-valued function 𝑟→𝐴^(𝑟) is continuous on [0,𝑟₀) for some 1>𝑟₀>0. Then the following assertions are equivalent:(1)𝐴∈𝐵_𝑝(ℓ²);(2)(1−𝑟²)²𝐴^(𝑟)∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²);(3)𝐴∈𝑃𝐿^𝑝(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(⋅))=^∞_𝑘=0(𝑘+1)¹⁰[𝐴(𝑠)]_𝑘𝑠^𝑘(2𝑠𝑑𝑠),(2.8) where 𝐴(⋅)∈𝐿^𝑝(𝐷,ℓ²). Then 𝑃1−𝑟²²𝐴_𝑘𝑟^𝑘(𝑠)=2𝑠^𝑘𝐴_𝑘(𝑘+2)(𝑘+3),(2.9) for all 𝑘≥0, and all 𝐴_𝑘∈𝐶_𝑝.
It follows that each matriceal polynomial is in 𝑃𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) for all 1≤𝑝<∞.
Suppose that 𝐴 is an upper triangular matrix with 𝐴_𝑘∈𝐶_𝑝 for all 𝑘≥0. We write 𝐴=⁴_𝑘=0𝐴_𝑘+𝐴¹,(2.10) where 𝐴¹∶=∑^∞𝑘=5𝐴_𝑘.
If (1−𝑟²)²𝐴^(𝑟)∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²), then we have that Φ(𝑟)∶=⁴_𝑘=0(𝑘+2)(𝑘+3)21−𝑟²²𝐴_𝑘𝑟^𝑘+1−𝑟²²𝐴¹^(𝑟)2!𝑟²(2.11) is in 𝐿^𝑝(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)‖‖^𝑝𝐶_𝑝𝑠^2𝑝𝑑𝑠<∞.(2.12) Consequently Φ∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²).
Moreover 𝐴=𝑃Φ since ^∞_𝑘=5¹⁰𝑘(𝑘−1)𝐴_𝑘𝑠^𝑘−21−𝑠²²𝑠²=(𝑘+1)𝑠^𝑘+1𝑑𝑠_∞_𝑘=5(𝑘−1)𝑘(𝑘+1)𝐴_𝑘¹⁰𝑠^2𝑘−31−𝑠²²𝑑𝑠=^∞_𝑘=5𝐴_𝑘.(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(⋅)∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²). Then we have that 1−𝑟²²𝐴^(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²‖‖𝐴^(𝑟)‖‖_𝐶12𝑟𝑑𝑟1−𝑟²²≤¹⁰¹⁰‖Φ(𝑠)‖_𝐶1^02𝜋6𝑠²𝑑𝜃||1−𝑟𝑠𝑒^𝑖𝜃||⁴=2𝑠𝑑𝑠2𝑟𝑑𝑟₁₀6𝑠²‖Φ(𝑠)‖_𝐶1¹⁰12𝜋^02𝜋𝑑𝜃||1−𝑟𝑠𝑒^𝑖𝜃||⁴∼2𝑟𝑑𝑟2𝑠𝑑𝑠₁₀‖Φ(𝑠)‖_𝐶112𝑠³1−𝑠²²𝑑𝑠≤6¹⁰‖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)=((𝑡_𝑖𝑗)(𝑟𝑠))^{∞𝑖,𝑗=1} be the Toeplitz matrix with 𝑡_𝑖𝑗(𝑟𝑠)=𝑡_𝑗−𝑖(𝑟𝑠)=𝑠²0(𝑟𝑠)^𝑗−𝑖(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)iofth𝑗e≥rw𝑖ise.(2.16) Since 𝑇(𝑟𝑠) is a Schur multiplier with ‖𝑇(𝑟𝑠)‖_𝑀(ℓ²₎=‖𝑇(𝑟𝑠)‖_𝐿¹_(𝕋)=‖6𝑠²/(1−𝑟𝑠𝑒^𝑖𝜃)⁴‖_𝐿¹_(𝕋) and 𝑀(ℓ²)=𝑀(𝐶₁)⊂𝑀(𝐶_𝑝), 1≤𝑝<∞ we get that 1−𝑟²²‖‖𝐴^(𝑟)‖‖_𝐶𝑝=1−𝑟²²‖‖‖¹⁰𝜙(𝑠)∗(1−6𝑠𝑟²𝑠)⁴(2𝑠𝑑𝑠)‖‖‖_𝐶𝑝≤1−𝑟²²¹⁰‖𝜙(𝑠)‖_𝐶𝑝‖‖‖‖6𝑠₂1−𝑟𝑠𝑒^𝑖𝜃⁴‖‖‖‖_𝐿1(𝕋)=1−𝑟₂(2𝑠𝑑𝑠)²¹⁰‖𝜙(𝑠)‖_𝐶𝑝1−𝑠²²‖‖‖‖6𝑠₂1−𝑟𝑠𝑒^𝑖𝜃⁴‖‖‖‖_𝐿1(𝕋)𝑑𝜆(𝑠)∶=𝑆_𝜙(𝑟).(2.17) From Schur’s theorem (see, e.g., [8]) it follows that 𝑆_𝜙(𝑟) is bounded on 𝐿^𝑝([0,1),𝑑𝜆) which in its turn implies that 1−𝑟²²𝐴^(𝑟)∈𝐿^𝑝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, [𝑉(𝐴=(⎧⎪⎨⎪⎩⋅)()𝑗](−𝑟)𝑖(+𝑖,𝑗3))(𝑗−𝑖+2)(𝑗−𝑖+1)2𝑟^𝑗−𝑖1−𝑟²²¹⁰0𝑎_𝑖𝑗(𝑠)𝑠^𝑗−𝑖(2𝑠𝑑𝑠)iofth𝑗e−rw𝑖i≥se.0,(3.1)
Then 𝑉 is an embedding from 𝐵_𝑝(ℓ²) into 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) for all 𝑝≥1, if 𝐵_𝑝(ℓ²)=𝑃𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) is equipped with the quotient norm.

Proof. Suppose that 𝐴∈𝐵_𝑝(ℓ²) and 𝐵(⋅)∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) with 𝐴=𝑃𝐵(⋅). Since 𝑃(𝐵(⋅))(𝑟)(𝑖,𝑗)=⎧⎪⎨⎪⎩2(𝑗−𝑖+1)𝑟^𝑗−𝑖¹⁰0𝑏_𝑖𝑗(𝑠)𝑠^{𝑗−𝑖+1}𝑑𝑠iofth𝑗e−rw𝑖i≥se,0(3.2) it is easy to see that 𝑃𝑉=𝑃,𝑉𝑃=𝑉(3.3) on 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²). Therefore 𝑉(𝐴)=𝑉(𝐵(⋅)) for all 𝐴∈𝐵_𝑝(ℓ²) and 𝐵(⋅)∈𝐿^𝑝(𝐷,𝑑𝜆,ℓ²).
We will now prove that 𝑉 is a bounded operator on𝐿^𝑝(𝐷,𝑑𝜆,ℓ²). We first prove this fact for 𝑝=1. By Fubini’s theorem we have that ‖𝑉(𝐴(⋅))‖_{𝐿1(𝐷,𝑑𝜆,ℓ}²₎==¹⁰‖[𝑉(𝐴(⋅))]‖_𝐶1𝑑𝜆(𝑟)₁₀‖‖‖‖^∞_𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟^𝑘1−𝑟²²¹⁰≤𝐴_𝑘(𝑠)𝑠^𝑘(2𝑠𝑑𝑠)‖‖‖‖_𝐶1𝑑𝜆(𝑟)₁₀‖‖‖‖^∞_𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟^𝑘(1−𝑟²)²𝐴_𝑘(𝑠)𝑠^𝑘+1‖‖‖‖_𝐶1=(2𝑑𝑠)𝑑𝜆(𝑟)₁₀⎡⎢⎢⎣¹⁰‖‖‖‖^∞_𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟^𝑘𝐴_𝑘(𝑠)𝑠^𝑘1−𝑠²²‖‖‖‖_𝐶1=(2𝑟𝑑𝑟)⎤⎥⎥⎦𝑑𝜆(𝑠)₁₀‖𝐴(𝑠)∗𝐶(𝑟𝑠)‖_𝐶1(2𝑟𝑑𝑟)𝑑𝜆(𝑠),(3.4) where 𝐶(𝑟𝑠)=(𝑐_𝑖𝑗(𝑟𝑠))^{∞𝑖,𝑙=1} means the Toeplitz matrix given by 𝑐_𝑖𝑗(𝑟𝑠)==𝑐(_𝑗−𝑖𝑟(𝑠𝑟)𝑠)_𝑗−𝑖𝑠²1−𝑠²²(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)02iofth𝑗e≥rw𝑖i,se.(3.5) Since the Toeplitz matrix 𝐶(𝑟𝑠) is a Schur multiplier with ‖𝐶(𝑟𝑠)‖_𝑀(ℓ²₎=‖‖‖‖6𝑠²1−𝑠²²1−𝑟𝑠𝑒^𝑖𝜃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≤¹⁰‖‖‖‖^∞_𝑘=0(𝑘+3)(𝑘+2)(𝑘+1)2𝑟^𝑘=𝑠^𝑘1−𝑟²²1−𝑠²²𝐴_𝑘(𝑠)‖‖‖‖_𝐶𝑝𝑑𝜆(𝑠)₁₀‖𝐴(𝑠)∗𝑇(𝑟𝑠)‖_𝐶𝑝𝑑𝜆(𝑠),(3.8) where 𝑇(𝑟𝑠)=(𝑡_𝑗−𝑖(𝑟𝑠))_𝑖,𝑗 is a Toeplitz matrix and 𝑡_𝑗−𝑖(𝑟𝑠)=(𝑟𝑠)^𝑗−𝑖1−𝑠²²1−𝑟²²(𝑗−𝑖+3)(𝑗−𝑖+2)(𝑗−𝑖+1)02iofth𝑗e≥rw𝑖ise.(3.9)𝑇(𝑟𝑠)is a Schur multiplier, therefore ¹⁰‖𝐴(𝑠)∗𝑇(𝑟𝑠)‖_𝐶𝑝𝑑𝜆(𝑠)≤¹⁰‖𝐴(𝑠)‖_𝐶𝑝1−𝑟²²1−𝑠²²‖‖‖‖61−𝑟𝑠𝑒^𝑖𝜃⁴‖‖‖‖_𝐿1(𝕋)∶=𝑆_𝐴(𝑟).(3.10) From Schur’s theorem, (see, e.g., [8]) we obtain that 𝑆_𝐴(𝑟) is bounded on 𝐿^𝑝([0,1),𝑑𝜆). Hence 𝑉 is bounded on 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²), 1≤𝑝<∞, and there is a constant 𝐶>0 such that ‖𝑉(𝐴(⋅))‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎≤𝐶‖𝐵(⋅)‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎(3.11) for all 𝐴=𝑃𝐵(⋅). Taking the infimum over 𝐵, we get that ‖𝑉(𝐴)‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎≤𝐶‖𝐴‖_𝐵𝑝(ℓ²₎.(3.12) Thus 𝑉∶𝐵_𝑝(ℓ²)→𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) is bounded.
On the other hand, since 𝑃𝑉=𝑃 and 𝑉𝑃=𝑉 on 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) we get easily that 𝐴=𝑃𝑉(𝐴) for all 𝐴∈𝐵_𝑝(ℓ²). Thus ‖𝐴‖_𝐵𝑝(ℓ²₎=inf‖𝐵(⋅)‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎∶𝐴=𝑃𝐵≤‖𝑉𝐴‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎,(3.13) and hence 𝑉∶𝐵_𝑝(ℓ²)→𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) is an embedding. The proof is complete.

We denote by ℬ_0,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⟩=¹⁰tr𝑉(𝐴)[𝑉(𝐵)]^∗𝑑𝜆(𝑟)(3.14) One has the following dualities:(1)𝐵_𝑝(ℓ²)^∗≈𝐵_𝑞(ℓ²) if 1<𝑝<∞ and 1/𝑝+1/𝑞=1;(2)ℬ_0,𝑐(𝐷,ℓ²)^∗≈𝐵₁(ℓ²) and 𝐵₁(ℓ²)^∗≈ℬ(𝐷,ℓ²).

Proof. Since 𝑉 is an embedding from 𝐵_𝑝(ℓ²) into 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²) for all 1≤𝑝<∞, Hölder’s inequality shows that 𝐵_𝑞(ℓ²)⊂𝐵_𝑝(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 𝐵_𝑝(ℓ²) with 1≤𝑝<∞. Then 𝐹∘𝑉⁻¹∶𝑉𝐵_𝑝(ℓ²)→ℂ extends to a bounded linear functional on 𝐿^𝑝(𝐷,𝑑𝜆,ℓ²). Thus there exists 𝐶(⋅)∈𝐿^𝑞(𝐷,𝑑𝜆,ℓ²) such that ‖𝐶(⋅)‖_{𝐿𝑞(𝐷,𝑑𝜆,ℓ}²₎=‖𝐹∘𝑉⁻¹‖ and 𝐹∘𝑉⁻¹(𝐵)=¹⁰tr(𝐵(𝑟))[𝐶(𝑟)]^∗𝑑𝜆(𝑟),𝐵(⋅)∈𝐿^𝑝𝐷,𝑑𝜆,ℓ².(3.15) In particular, if 𝐵(⋅)=𝑉(𝐴) with 𝐴∈𝐵_𝑝(ℓ²), then 𝐹(𝐴)=¹⁰tr((𝑉𝐴)(𝑟))[𝐶(𝑟)]^∗𝑑𝜆(𝑟).(3.16) Let 𝐵=𝑃(𝐶). Then 𝐵∈𝐵_𝑞(ℓ²) and it is easy to check that 𝐹(𝐴)=¹⁰tr(𝑉𝐴)(𝑟)[(𝑉𝐵)(𝑟)]^∗𝑑𝜆(𝑟),𝐴∈𝐵_𝑝ℓ²,(3.17) with ‖𝐵‖_𝐵𝑞(ℓ²₎≤‖𝐶(⋅)‖_{𝐿𝑝(𝐷,𝑑𝜆,ℓ}²₎=‖𝐹∘𝑉⁻¹‖≤‖𝑉⁻¹‖‖𝐹‖. This proves the duality 𝐵_𝑝(ℓ²)^∗≈𝐵_𝑞(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 ℬ_0,𝑐(𝐷,ℓ²). Then we will prove that there is a matrix 𝐶 from 𝐵₁(ℓ²) such that 𝐹(𝐵)=¹⁰tr𝑉𝐵(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)→𝒞₀(𝐷,ℓ²) is an isomorphic embedding. Thus 𝑋=𝑉(ℬ_0,𝑐(𝐷,ℓ²)) is a closed subspace in 𝒞₀(𝐷,𝐶_∞) and 𝐹∘(𝑉)⁻¹∶𝑋→ℂ is a bounded linear functional on 𝑋, where ℂ₀(𝐷,𝐶_∞) is the subset in 𝒞₀(𝐷,ℓ²) 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 Φ∶𝒞₀(𝐷,𝐶_∞)→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]×𝑈_𝐶1, where 𝑈_𝐶1 is the unit ball of the space 𝐶₁ with the topology 𝜎(𝐶₁,𝐶_∞), such that Φ(𝑓⊗𝐴)=_{[0,1]×𝑈𝐶1}𝑓(𝑟)tr𝐴𝐵^∗𝑑𝜇(𝑟,𝐵)(3.19) for every 𝑓∈𝒞₀[0,1] and 𝐴∈𝐶_∞.
Thus, for the matrix ∑^𝑛𝑘=0𝐴_𝑘∈ℬ_0,𝑐(𝐷,ℓ²), identified with the analytic matrix ∑^𝑛𝑘=0𝐴_𝑘𝑟^𝑘, we have that 𝐹^𝑛_𝑘=0𝐴_𝑘=𝐹^𝑛_𝑘=0𝑟^𝑘𝐴_𝑘=𝐹∘(𝑉)⁻¹𝑉^𝑛_𝑘=0𝑟^𝑘𝐴_𝑘=Φ_𝑛_𝑘=0(𝑘+3)(𝑘+2)2𝑟^𝑘1−𝑟²²𝐴_𝑘=_{[0,1]×𝑈𝐶1𝑛}_𝑘=0tr(𝑘+3)(𝑘+2)2𝑟^𝑘1−𝑟²²𝐴_𝑘𝐵^∗𝑑𝜇(𝑟,𝐵)_def=𝜇(𝑟,𝐵),tr^𝑛_𝑘=0(𝑘+3)(𝑘+2)2𝑟^𝑘𝐴_𝑘𝐵^∗1−𝑟²².(3.20) On the other hand, we wish to have that 𝐹(𝐴)==¹⁰tr𝑉(𝐴)(𝑉(𝐶))^∗𝑑𝜆(𝑠)₁₀tr^𝑛_𝑘=0(𝑘+3)(𝑘+2)2𝑠^𝑘𝐴_𝑘(𝑉(𝐶))^∗=(2𝑠𝑑𝑠)₁₀tr^𝑛_𝑘=0𝑠^2𝑘(𝑘+3)²(𝑘+2)²41−𝑠²²𝐴_𝑘𝐶^∗𝑘=(2𝑠𝑑𝑠)_𝑛_𝑘=0tr𝐴_𝑘(𝑘+2(3𝑘)+(𝑘1+)2)𝐶^∗𝑘.(3.21) Therefore, letting 𝐴=𝑒_{𝑖,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 𝐶_𝑘=𝜇(𝑟,𝐵),(𝑘+1)𝑟^𝑘1−𝑟²²𝐵_𝑘,𝑘=0,1,2,….(3.22) Then, it yields that ¹⁰‖‖𝐶=^(𝑠)‖‖_𝐶12𝑠𝑑𝑠₁₀‖‖‖‖_{[0,1]×𝑈𝐶1}^𝑛_𝑘=2(𝑘+1)!(𝑘−2)!𝑠^𝑘−2≤𝑟^𝑘1−𝑟²²𝐵_𝑘𝑑𝜇(𝑟,𝐵)‖‖‖‖_𝐶1(2𝑠𝑑𝑠)_{[0,1]×𝑈𝐶1}⎡⎢⎢⎣¹⁰‖‖‖‖^𝑛_𝑘=2(𝑘+1)!(𝑘−2)!(𝑟𝑠)^𝑘−2𝑟²1−𝑟²²𝐵_𝑘‖‖‖‖_𝐶1≤(2𝑠𝑑𝑠)⎤⎥⎥⎦𝑑||𝜇||(𝑟,𝐵)_{[0,1]×𝑈𝐶1}⎡⎢⎢⎣¹⁰‖‖‖‖^𝑛_𝑘=2(𝑘+1)!(𝑘−2)!(𝑟𝑠)^𝑘−2𝑟²1−𝑟²²𝑒^{𝑖𝑘(⋅)}‖‖‖‖_𝐿1(𝕋)≤‖𝐵‖_𝐶1(2𝑠𝑑𝑠)⎤⎥⎥⎦𝑑||𝜇||(𝑟,𝐵)_{[0,1]×𝑈𝐶1}¹⁰^02𝜋𝑟²1−𝑟²²||1−𝑟𝑠𝑒^𝑖𝜃||⁴𝑑𝜃∼2𝜋(2𝑠𝑑𝑠)𝑑||𝜇||(𝑟,𝐵)_{[0,1]×𝑈𝐶1}𝑟²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=0𝐴_𝑘 is dense in ℬ_0,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))=𝐻^11,∞,1(ℓ²)=^def{𝐴∶sup_𝑟<1(1−𝑟)‖∑_𝑘≥0𝑘𝐴_𝑘‖_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.

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.