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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 693251, 13 pages
http://dx.doi.org/10.1155/2012/693251
Research Article

Besov-Schatten Spaces

1Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
2Department of Mathematics, LuleΓ₯ University of Technology, 97 187 LuleΓ₯, Sweden

Received 22 March 2011; Accepted 9 April 2011

Academic Editor: NicolaeΒ Popa

Copyright Β© 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.

Abstract

We introduce the Besov-Schatten spaces 𝐡𝑝(β„“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 𝐡1(β„“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œ‹-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=(π‘Žπ‘—π‘˜)𝑗,π‘˜β‰₯0 and 𝐡=(π‘π‘—π‘˜)𝑗,π‘˜β‰₯0 is defined as the matrix π΄βˆ—π΅ whose entries are the products of the entries of 𝐴 and 𝐡: ξ€·π‘Žπ΄βˆ—π΅=π‘—π‘˜π‘π‘—π‘˜ξ€Έπ‘—,π‘˜β‰₯0.(1.1) If 𝑋 and π‘Œ are two Banach spaces of matrices we define Schur multipliers from 𝑋 to π‘Œ as the space 𝑀(𝑋,π‘Œ)={π‘€βˆΆπ‘€βˆ—π΄βˆˆπ‘Œforeveryπ΄βˆˆπ‘‹},(1.2) equipped with the natural norm ‖𝑀‖=sup‖𝐴‖𝑋≀1β€–π‘€βˆ—π΄β€–π‘Œ.(1.3) In the case 𝑋=π‘Œ=𝐡(β„“2), where 𝐡(β„“2) is the space of all linear and bounded operators on β„“2, the space 𝑀(𝐡(β„“2),𝐡(β„“2)) will be denoted 𝑀(β„“2) and a matrix π΄βˆˆπ‘€(β„“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=(π‘π‘—βˆ’π‘˜)𝑗,π‘˜, where (𝑐𝑛)π‘›βˆˆβ„€ is a sequence of complex numbers, is a Schur multiplier if and only if there exists a bounded and complex Borel measure πœ‡ on (the circle group) 𝕋 with ξπœ‡(𝑛)=𝑐𝑛,for𝑛=0,Β±1,Β±2,….(1.4) Moreover, one then has that ‖𝑀‖=β€–πœ‡β€–.(1.5)

We will denote by 𝐢𝑝, 0<𝑝<∞, the Schatten class operators (see, e.g., [8]). Let us summarize briefly some well-known properties of classes 𝑀(𝐢𝑝) which will be very often used in what follows. If 1<𝑝<∞, then 𝑀(𝐢𝑝)=𝑀(πΆπ‘ξ…ž), where 1/𝑝+1/𝑝′=1 and 𝑀(β„“2)=𝑀(𝐢1). Next, interpolating between the classes 𝐢𝑝, we can easily see that 𝑀(𝐢𝑝1)βŠ‚π‘€(𝐢𝑝2) if 0<𝑝1≀𝑝2≀2 (see, e.g., [9]). We will denote by π΄π‘˜, the π‘˜th-diagonal matrix associated to 𝐴 (see [4]). For an infinite matrix 𝐴=(π‘Žπ‘–π‘—) and an integer π‘˜ we denote by π΄π‘˜ the matrix whose entries π‘Žβ€²π‘–π‘— are given by π‘Žβ€²π‘–π‘—=ξ‚»π‘Žπ‘–π‘—ifπ‘—βˆ’π‘–=π‘˜,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(π‘Ÿ). These functions may be regarded as infinite matrix-valued functions defined on the unit disc 𝐷 using the correspondence 𝐴(π‘Ÿ)βŸΆπ‘“π΄(π‘Ÿ,𝑑)=βˆžξ“π‘˜=βˆ’βˆžπ΄π‘˜(π‘Ÿ)π‘’π‘–π‘˜π‘‘,(1.7) where π΄π‘˜(π‘Ÿ) is the π‘˜th-diagonal of the matrix 𝐴(π‘Ÿ), the preceding sum is a formal one, and 𝑑 belongs to the torus 𝕋. This matrix 𝐴(π‘Ÿ) is called analytic matrix if there exists an upper triangular infinite matrix 𝐴 such that, for all π‘Ÿβˆˆ[0,1), we have π΄π‘˜(π‘Ÿ)=π΄π‘˜π‘Ÿπ‘˜, for all π‘˜βˆˆβ„€. In what follows we identify the analytic matrices 𝐴(π‘Ÿ) 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,β„“2) is the space of all analytic matrices 𝐴 with 𝐴(π‘Ÿ)∈𝐡(β„“2), 0β‰€π‘Ÿ<1, such that ‖𝐴‖ℬ(𝐷,β„“2)=sup0β‰€π‘Ÿ<1ξ€·1βˆ’π‘Ÿ2ξ€Έβ€–β€–π΄ξ…žβ€–β€–(π‘Ÿ)𝐡(β„“2)+‖‖𝐴0‖‖𝐡(β„“2)<∞,(1.8) where 𝐡(β„“2) is the usual operator norm of the matrix 𝐴 on the sequence space β„“2 andβˆ‘π΄β€²(π‘Ÿ)=βˆžπ‘˜=0π΄π‘˜π‘˜π‘Ÿπ‘˜βˆ’1.

The space ℬ0(𝐷,β„“2) is the space of all upper triangular infinite matrices 𝐴 such that limπ‘Ÿβ†’1βˆ’(1βˆ’π‘Ÿ2)β€–(π΄βˆ—πΆ(π‘Ÿ))′‖𝐡(β„“2)=0, where 𝐢(π‘Ÿ) is the Toeplitz matrix associated with the Cauchy kernel 1/(1βˆ’π‘Ÿ), for 0β‰€π‘Ÿ<1.

An important tool in this paper is the Bergman projection. It is known (see, e.g., [10]) that for all strong measurable 𝐢𝑝-valued functions π‘Ÿβ†’π΄(π‘Ÿ) defined on [0,1) with ∫10‖𝐴(π‘Ÿ)‖𝑝𝐢𝑝2π‘Ÿπ‘‘π‘Ÿ<∞ and for all 𝑖,π‘—βˆˆβ„• we have that []⎧βŽͺ⎨βŽͺβŽ©π‘ƒ(𝐴(β‹…))(π‘Ÿ)(𝑖,𝑗)=2(π‘—βˆ’π‘–+1)π‘Ÿπ‘—βˆ’π‘–ξ€œ10π‘Žπ‘–π‘—(𝑠)β‹…π‘ π‘—βˆ’π‘–+1𝑑𝑠,if𝑖≀𝑗,0,otherwise.(1.9)

Now we consider a modified version of Bergman projection.

Let 𝛼>βˆ’1. Thenξ€Ίπ‘ƒπ›Όξ€»βŽ§βŽͺ⎨βŽͺ⎩𝐴(β‹…)(π‘Ÿ)=(𝛼+1)Ξ“(π‘—βˆ’π‘–+2+𝛼)π‘Ÿ(π‘—βˆ’π‘–)!Ξ“(𝛼+2)π‘—βˆ’π‘–ξ‚΅2ξ€œ10π‘Žπ‘–π‘—(𝑠)π‘ π‘—βˆ’π‘–+1ξ€·1βˆ’π‘ 2𝛼𝑑𝑠if𝑗β‰₯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)βˆ—, that is, 𝑃2𝐴(β‹…)βˆ—=⎧βŽͺ⎨βŽͺ⎩(π‘Ÿ)(𝑖,𝑗)(π‘—βˆ’π‘–+3)(π‘—βˆ’π‘–+2)(π‘—βˆ’π‘–+1)2π‘Ÿπ‘—βˆ’π‘–ξ€·1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10π‘Žπ‘–π‘—(𝑠)π‘ π‘—βˆ’π‘–(2𝑠𝑑𝑠)ifπ‘—βˆ’π‘–β‰₯00otherwise.(1.11)
Then 𝑉 is an isomorphic embedding of ℬ0(𝐷,β„“2) in π’ž0(𝐷,β„“2), where π’ž0(𝐷,β„“2) is the space of all continuous 𝐡(β„“2)-valued functions 𝐡(π‘Ÿ) on [0,1) such that limπ‘Ÿβ†’1𝐡(π‘Ÿ)=0 in the norm of 𝐡(β„“2).

The paper is organized as follows. In Section 2 we give a characterization of matrices in the Besov-Schatten space 𝐡𝑝(β„“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 1≀𝑝<∞ and a positive measure on [0,1) given by π‘‘πœ†(π‘Ÿ)=2π‘Ÿπ‘‘π‘Ÿξ€·1βˆ’π‘Ÿ2ξ€Έ2.(2.1) The Besov-Schatten matrix space 𝐡𝑝(β„“2) is defined to be the space of all upper triangular infinite matrices 𝐴 such that ‖𝐴‖𝐡𝑝(β„“2)=ξ‚Έξ€œ10ξ€·1βˆ’π‘Ÿ2ξ€Έ2π‘β€–β€–π΄ξ…žξ…žβ€–β€–(π‘Ÿ)π‘πΆπ‘ξ‚Ήπ‘‘πœ†(π‘Ÿ)1/𝑝<∞.(2.2)

On 𝐡𝑝(β„“2) we introduce the norm ‖‖𝐴‖𝐴‖=0‖‖𝐢1+‖𝐴‖𝐡𝑝(β„“2).(2.3)

We introduce the notation 𝐿𝑝(𝐷,π‘‘πœ†,β„“2) for the space of all strongly measurable functions π‘Ÿβ†’π΄(π‘Ÿ) defined on the measurable space ([0,1),π‘‘πœ†) with 𝐢𝑝 values such that ‖𝐴‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2)=ξ‚΅ξ€œ10‖𝐴(π‘Ÿ)β€–π‘πΆπ‘ξ‚Άπ‘‘πœ†(π‘Ÿ)1/𝑝<∞.(2.4)

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

Lemma 2.2. Let π‘§βˆˆπ·, 𝑐 is real, 𝑑>βˆ’1, and 𝐼𝑐,𝑑=ξ€œπ·ξ€·1βˆ’|𝑀|2𝑑||1βˆ’π‘§π‘€||2+𝑑+𝑐𝑑𝐴(𝑀).(2.5) Then,(1)if 𝑐<0, then 𝐼𝑐,𝑑(𝑧) is bounded in 𝑧;(2)if 𝑐>0, then 𝐼𝑐,𝑑1(𝑧)βˆΌξ€·1βˆ’|𝑧|2𝑐(|𝑧|⟢1βˆ’);(2.6)(3)if 𝑐=0, then 𝐼0,𝑐1(𝑧)∼log1βˆ’|𝑧|2(|𝑧|⟢1βˆ’).(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 𝐢𝑝-valued function π‘Ÿβ†’π΄ξ…žξ…ž(π‘Ÿ) is continuous on [0,π‘Ÿ0) for some 1>π‘Ÿ0>0. Then the following assertions are equivalent:(1)π΄βˆˆπ΅π‘(β„“2);(2)(1βˆ’π‘Ÿ2)2π΄ξ…žξ…ž(π‘Ÿ)βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2);(3)π΄βˆˆπ‘ƒπΏπ‘(𝐷,π‘‘πœ†,β„“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(β‹…))=βˆžξ“π‘˜=0(ξ€œπ‘˜+1)10[]𝐴(𝑠)π‘˜π‘ π‘˜(2𝑠𝑑𝑠),(2.8) where 𝐴(β‹…)βˆˆπΏπ‘(𝐷,β„“2). Then 𝑃1βˆ’π‘Ÿ2ξ€Έ2π΄π‘˜π‘Ÿπ‘˜ξ‚(𝑠)=2π‘ π‘˜π΄π‘˜,(π‘˜+2)(π‘˜+3)(2.9) for all π‘˜β‰₯0, and all π΄π‘˜βˆˆπΆπ‘.
It follows that each matriceal polynomial is in 𝑃𝐿𝑝(𝐷,π‘‘πœ†,β„“2) for all 1≀𝑝<∞.
Suppose that 𝐴 is an upper triangular matrix with π΄π‘˜βˆˆπΆπ‘ for all π‘˜β‰₯0. We write 𝐴=4ξ“π‘˜=0π΄π‘˜+𝐴1,(2.10) where 𝐴1βˆ‘βˆΆ=βˆžπ‘˜=5π΄π‘˜.
If (1βˆ’π‘Ÿ2)2π΄ξ…žξ…ž(π‘Ÿ)βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2), then we have that Ξ¦(π‘Ÿ)∢=4ξ“π‘˜=0(π‘˜+2)(π‘˜+3)2ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΄π‘˜π‘Ÿπ‘˜+ξ€·1βˆ’π‘Ÿ2ξ€Έ2𝐴1ξ€Έξ…žξ…ž(π‘Ÿ)2!π‘Ÿ2(2.11) is in 𝐿𝑝(𝐷,π‘‘πœ†,β„“2) and moreover that 𝐴=𝑃Φ.
Indeed, for 0<π‘Ÿ<π‘Ÿ0, π‘Ÿβ†’(𝐴1)ξ…žξ…ž(π‘Ÿ) is a continuous function and, therefore ξ€œπ‘Ÿ00‖‖𝐴1ξ€Έξ…žξ…žβ€–β€–(𝑠)𝑝𝐢𝑝𝑠2𝑝𝑑𝑠<∞.(2.12) Consequently Ξ¦βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2).
Moreover 𝐴=𝑃Φ since βˆžξ“π‘˜=5ξ€œ10π‘˜(π‘˜βˆ’1)π΄π‘˜π‘ π‘˜βˆ’2ξ€·1βˆ’π‘ 2ξ€Έ2𝑠2(π‘˜+1)π‘ π‘˜+1=π‘‘π‘ βˆžξ“π‘˜=5(π‘˜βˆ’1)π‘˜(π‘˜+1)π΄π‘˜ξ€œ10𝑠2π‘˜βˆ’3ξ€·1βˆ’π‘ 2ξ€Έ2𝑑𝑠=βˆžξ“π‘˜=5π΄π‘˜.(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 Ξ¦(β‹…)βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2). Then we have that ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΄ξ…žξ…žξ€·(π‘Ÿ)=1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10ξ‚ΈΞ¦(𝑠)βˆ—6𝑠2(1βˆ’π‘Ÿπ‘ )4ξ‚Ή2𝑠𝑑𝑠.(2.14) Using Fubini’s theorem and Lemma 2.2 we obtain that ξ€œ10ξ€·1βˆ’π‘Ÿ2ξ€Έ2β€–β€–π΄ξ…žξ…žβ€–β€–(π‘Ÿ)𝐢12π‘Ÿπ‘‘π‘Ÿξ€·1βˆ’π‘Ÿ2ξ€Έ2β‰€ξ€œ10ξƒ¬ξ€œ10β€–Ξ¦(𝑠)‖𝐢1ξ€œ02πœ‹6𝑠2π‘‘πœƒ||1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒ||4ξƒ­=ξ€œ2𝑠𝑑𝑠2π‘Ÿπ‘‘π‘Ÿ106𝑠2(‖Φ𝑠)‖𝐢1ξƒ¬ξ€œ101ξ€œ2πœ‹02πœ‹π‘‘πœƒ||1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒ||4ξƒ­βˆΌξ€œ2π‘Ÿπ‘‘π‘Ÿ2𝑠𝑑𝑠10β€–Ξ¦(𝑠)‖𝐢112𝑠3ξ€·1βˆ’π‘ 2ξ€Έ2ξ€œπ‘‘π‘ β‰€610β€–Ξ¦(𝑠)‖𝐢1π‘‘πœ†(𝑠)<∞.(2.15)
Consequently, 𝐴∈𝐿1(𝐷,π‘‘πœ†,β„“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(π‘Ÿπ‘ )=((𝑑𝑖𝑗)(π‘Ÿπ‘ ))βˆžπ‘–,𝑗=1 be the Toeplitz matrix with 𝑑𝑖𝑗(π‘Ÿπ‘ )=π‘‘π‘—βˆ’π‘–ξ‚»π‘ (π‘Ÿπ‘ )=2(π‘Ÿπ‘ )π‘—βˆ’π‘–(π‘—βˆ’π‘–+3)(π‘—βˆ’π‘–+2)(π‘—βˆ’π‘–+1)if𝑗β‰₯𝑖0otherwise.(2.16) Since 𝑇(π‘Ÿπ‘ ) is a Schur multiplier with ‖𝑇(π‘Ÿπ‘ )‖𝑀(β„“2)=‖𝑇(π‘Ÿπ‘ )‖𝐿1(𝕋)=β€–6𝑠2/(1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒ)4‖𝐿1(𝕋) and 𝑀(β„“2)=𝑀(𝐢1)βŠ‚π‘€(𝐢𝑝), 1≀𝑝<∞ we get that ξ€·1βˆ’π‘Ÿ2ξ€Έ2β€–β€–π΄ξ…žξ…žβ€–β€–(π‘Ÿ)𝐢𝑝=ξ€·1βˆ’π‘Ÿ2ξ€Έ2β€–β€–β€–ξ€œ10ξ‚Έπœ™(𝑠)βˆ—6𝑠2(1βˆ’π‘Ÿπ‘ )4ξ‚Ήβ€–β€–β€–(2𝑠𝑑𝑠)𝐢𝑝≀1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10β€–β€–πœ™(𝑠)𝐢𝑝‖‖‖‖6𝑠2ξ€·1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒξ€Έ4‖‖‖‖𝐿1(𝕋)=ξ€·(2𝑠𝑑𝑠)1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10β€–β€–πœ™(𝑠)𝐢𝑝1βˆ’π‘ 2ξ€Έ2β€–β€–β€–β€–6𝑠2ξ€·1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒξ€Έ4‖‖‖‖𝐿1(𝕋)π‘‘πœ†(𝑠)∢=π‘†πœ™(π‘Ÿ).(2.17) From Schur’s theorem (see, e.g., [8]) it follows that π‘†πœ™(π‘Ÿ) is bounded on 𝐿𝑝([0,1),π‘‘πœ†) which in its turn implies that ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΄ξ…žξ…ž(π‘Ÿ)βˆˆπΏπ‘ξ€·π·,π‘‘πœ†,β„“2ξ€Έ(2.18) for 1<𝑝<∞. Thus also the implication (3)β‡’(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)βˆ—, that is, []=⎧βŽͺ⎨βŽͺβŽ©π‘‰(𝐴(β‹…))(π‘Ÿ)(𝑖,𝑗)(π‘—βˆ’π‘–+3)(π‘—βˆ’π‘–+2)(π‘—βˆ’π‘–+1)2π‘Ÿπ‘—βˆ’π‘–ξ€·1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10π‘Žπ‘–π‘—(𝑠)π‘ π‘—βˆ’π‘–(2𝑠𝑑𝑠)ifπ‘—βˆ’π‘–β‰₯0,0otherwise.(3.1)
Then 𝑉 is an embedding from 𝐡𝑝(β„“2) into 𝐿𝑝(𝐷,π‘‘πœ†,β„“2) for all 𝑝β‰₯1, if 𝐡𝑝(β„“2)=𝑃𝐿𝑝(𝐷,π‘‘πœ†,β„“2) is equipped with the quotient norm.

Proof. Suppose that π΄βˆˆπ΅π‘(β„“2) and 𝐡(β‹…)βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2) with 𝐴=𝑃𝐡(β‹…). Since ⎧βŽͺ⎨βŽͺβŽ©π‘ƒ(𝐡(β‹…))(π‘Ÿ)(𝑖,𝑗)=2(π‘—βˆ’π‘–+1)π‘Ÿπ‘—βˆ’π‘–ξ€œ10𝑏𝑖𝑗(𝑠)π‘ π‘—βˆ’π‘–+1𝑑𝑠ifπ‘—βˆ’π‘–β‰₯00otherwise,(3.2) it is easy to see that 𝑃𝑉=𝑃,𝑉𝑃=𝑉(3.3) on 𝐿𝑝(𝐷,π‘‘πœ†,β„“2). Therefore 𝑉(𝐴)=𝑉(𝐡(β‹…)) for all π΄βˆˆπ΅π‘(β„“2) and 𝐡(β‹…)βˆˆπΏπ‘(𝐷,π‘‘πœ†,β„“2).
We will now prove that 𝑉 is a bounded operator on𝐿𝑝(𝐷,π‘‘πœ†,β„“2). We first prove this fact for 𝑝=1. By Fubini’s theorem we have that ‖‖𝑉(𝐴(β‹…))𝐿1(𝐷,π‘‘πœ†,β„“2)=ξ€œ10β€–[]‖𝑉(𝐴(β‹…))𝐢1=ξ€œπ‘‘πœ†(π‘Ÿ)10β€–β€–β€–β€–βˆžξ“π‘˜=0(π‘˜+3)(π‘˜+2)(π‘˜+1)2π‘Ÿπ‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2ξ€œ10π΄π‘˜(𝑠)π‘ π‘˜(β€–β€–β€–β€–2𝑠𝑑𝑠)𝐢1β‰€ξ€π‘‘πœ†(π‘Ÿ)10β€–β€–β€–β€–βˆžξ“π‘˜=0(π‘˜+3)(π‘˜+2)(π‘˜+1)2π‘Ÿπ‘˜(1βˆ’π‘Ÿ2)2π΄π‘˜(𝑠)π‘ π‘˜+1‖‖‖‖𝐢1=ξ€œ(2𝑑𝑠)π‘‘πœ†(π‘Ÿ)10βŽ‘βŽ’βŽ’βŽ£ξ€œ10β€–β€–β€–β€–βˆžξ“π‘˜=0(π‘˜+3)(π‘˜+2)(π‘˜+1)2π‘Ÿπ‘˜π΄π‘˜(𝑠)π‘ π‘˜ξ€·1βˆ’π‘ 2ξ€Έ2‖‖‖‖𝐢1⎀βŽ₯βŽ₯⎦=(2π‘Ÿπ‘‘π‘Ÿ)π‘‘πœ†(𝑠)10‖𝐴(𝑠)βˆ—πΆ(π‘Ÿπ‘ )‖𝐢1(2π‘Ÿπ‘‘π‘Ÿ)π‘‘πœ†(𝑠),(3.4) where 𝐢(π‘Ÿπ‘ )=(𝑐𝑖𝑗(π‘Ÿπ‘ ))βˆžπ‘–,𝑙=1 means the Toeplitz matrix given by 𝑐𝑖𝑗(π‘Ÿπ‘ )=π‘π‘—βˆ’π‘–=ξƒ―((π‘Ÿπ‘ )π‘Ÿπ‘ )π‘—βˆ’π‘–π‘ 2ξ€·1βˆ’π‘ 2ξ€Έ2(π‘—βˆ’π‘–+3)(π‘—βˆ’π‘–+2)(π‘—βˆ’π‘–+1)2if𝑗β‰₯𝑖,0otherwise.(3.5) Since the Toeplitz matrix 𝐢(π‘Ÿπ‘ ) is a Schur multiplier with ‖𝐢(π‘Ÿπ‘ )‖𝑀(β„“2)=β€–β€–β€–β€–6𝑠2ξ€·1βˆ’π‘ 2ξ€Έ2ξ€·1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒξ€Έ4‖‖‖‖𝐿1(𝕋),(3.6) then, according to Lemma 2.2, it follows that 10‖‖𝐴(𝑠)βˆ—πΆ(π‘Ÿπ‘ )𝐢1ξ€œ(2π‘Ÿπ‘‘π‘Ÿ)π‘‘πœ†(𝑠)≀10‖‖𝐴(𝑠)𝐢1ξ€œ10‖‖𝐢(π‘Ÿπ‘ )𝑀(β„“2)βˆΌξ€œ(2π‘Ÿπ‘‘π‘Ÿ)π‘‘πœ†(𝑠)10‖‖𝐴(𝑠)𝐢1π‘‘πœ†(𝑠).(3.7) Consequently 𝑉 is bounded on 𝐿1(𝐷,π‘‘πœ†,β„“2). For 1<𝑝<∞ we have that (‖𝑉𝐴⋅)(π‘Ÿ)β€–πΆπ‘β‰€ξ€œ10β€–β€–β€–β€–βˆžξ“π‘˜=0(π‘˜+3)(π‘˜+2)(π‘˜+1)2π‘Ÿπ‘˜π‘ π‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2ξ€·1βˆ’π‘ 2ξ€Έ2π΄π‘˜(‖‖‖‖𝑠)𝐢𝑝=ξ€œπ‘‘πœ†(𝑠)10(‖𝐴𝑠)βˆ—π‘‡(π‘Ÿπ‘ )β€–πΆπ‘π‘‘πœ†(𝑠),(3.8) where 𝑇(π‘Ÿπ‘ )=(π‘‘π‘—βˆ’π‘–(π‘Ÿπ‘ ))𝑖,𝑗 is a Toeplitz matrix and π‘‘π‘—βˆ’π‘–ξƒ―((π‘Ÿπ‘ )=π‘Ÿπ‘ )π‘—βˆ’π‘–ξ€·1βˆ’π‘ 2ξ€Έ2ξ€·1βˆ’π‘Ÿ2ξ€Έ2(π‘—βˆ’π‘–+3)(π‘—βˆ’π‘–+2)(π‘—βˆ’π‘–+1)2if𝑗β‰₯𝑖0otherwise.(3.9)𝑇(π‘Ÿπ‘ )is a Schur multiplier, therefore ξ€œ10‖𝐴(𝑠)βˆ—π‘‡(π‘Ÿπ‘ )β€–πΆπ‘ξ€œπ‘‘πœ†(𝑠)≀10‖𝐴(𝑠)‖𝐢𝑝1βˆ’π‘Ÿ2ξ€Έ2ξ€·1βˆ’π‘ 2ξ€Έ2β€–β€–β€–β€–6ξ€·1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒξ€Έ4‖‖‖‖𝐿1(𝕋)∢=𝑆𝐴(π‘Ÿ).(3.10) From Schur’s theorem, (see, e.g., [8]) we obtain that 𝑆𝐴(π‘Ÿ) is bounded on 𝐿𝑝([0,1),π‘‘πœ†). Hence 𝑉 is bounded on 𝐿𝑝(𝐷,π‘‘πœ†,β„“2), 1≀𝑝<∞, and there is a constant 𝐢>0 such that ‖𝑉(𝐴(β‹…))‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2)≀𝐢‖𝐡(β‹…)‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2)(3.11) for all 𝐴=𝑃𝐡(β‹…). Taking the infimum over 𝐡, we get that ‖𝑉(𝐴)‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2)≀𝐢‖𝐴‖𝐡𝑝(β„“2).(3.12) Thus π‘‰βˆΆπ΅π‘(β„“2)→𝐿𝑝(𝐷,π‘‘πœ†,β„“2) is bounded.
On the other hand, since 𝑃𝑉=𝑃 and 𝑉𝑃=𝑉 on 𝐿𝑝(𝐷,π‘‘πœ†,β„“2) we get easily that 𝐴=𝑃𝑉(𝐴) for all π΄βˆˆπ΅π‘(β„“2). Thus ‖𝐴‖𝐡𝑝(β„“2)ξ€½β€–=inf‖𝐡(β‹…)𝐿𝑝(𝐷,π‘‘πœ†,β„“2)ξ€ΎβˆΆπ΄=𝑃𝐡≀‖𝑉𝐴‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2),(3.13) and hence π‘‰βˆΆπ΅π‘(β„“2)→𝐿𝑝(𝐷,π‘‘πœ†,β„“2) is an embedding. The proof is complete.

We denote by ℬ0,𝑐(𝐷,β„“2) the closed Banach subspace of ℬ0(𝐷,β„“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 ξ€œβŸ¨π΄,𝐡⟩=10ξ€·[]tr𝑉(𝐴)𝑉(𝐡)βˆ—ξ€Έπ‘‘πœ†(π‘Ÿ)(3.14) One has the following dualities:(1)𝐡𝑝(β„“2)βˆ—β‰ˆπ΅π‘ž(β„“2) if 1<𝑝<∞ and 1/𝑝+1/π‘ž=1;(2)ℬ0,𝑐(𝐷,β„“2)βˆ—β‰ˆπ΅1(β„“2) and 𝐡1(β„“2)βˆ—β‰ˆβ„¬(𝐷,β„“2).

Proof. Since 𝑉 is an embedding from 𝐡𝑝(β„“2) into 𝐿𝑝(𝐷,π‘‘πœ†,β„“2) for all 1≀𝑝<∞, HΓΆlder’s inequality shows that π΅π‘ž(β„“2)βŠ‚π΅π‘(β„“2)βˆ— for 1≀𝑝<∞ and 𝐡1(β„“2)βŠ‚β„¬βˆ—0,𝑐(𝐷,β„“2).
Suppose that 𝐹 is a bounded linear functional on the Besov-Schatten space 𝐡𝑝(β„“2) with 1≀𝑝<∞. Then πΉβˆ˜π‘‰βˆ’1βˆΆπ‘‰π΅π‘(β„“2)β†’β„‚ extends to a bounded linear functional on 𝐿𝑝(𝐷,π‘‘πœ†,β„“2). Thus there exists 𝐢(β‹…)βˆˆπΏπ‘ž(𝐷,π‘‘πœ†,β„“2) such that ‖𝐢(β‹…)β€–πΏπ‘ž(𝐷,π‘‘πœ†,β„“2)=β€–πΉβˆ˜π‘‰βˆ’1β€– and ξ€·πΉβˆ˜π‘‰βˆ’1ξ€Έξ€œ(𝐡)=10[]tr(𝐡(π‘Ÿ))𝐢(π‘Ÿ)βˆ—π‘‘πœ†(π‘Ÿ),𝐡(β‹…)βˆˆπΏπ‘ξ€·π·,π‘‘πœ†,β„“2ξ€Έ.(3.15) In particular, if 𝐡(β‹…)=𝑉(𝐴) with π΄βˆˆπ΅π‘(β„“2), then ξ€œπΉ(𝐴)=10[]tr((𝑉𝐴)(π‘Ÿ))𝐢(π‘Ÿ)βˆ—π‘‘πœ†(π‘Ÿ).(3.16) Let 𝐡=𝑃(𝐢). Then π΅βˆˆπ΅π‘ž(β„“2) and it is easy to check that ξ€œπΉ(𝐴)=10ξ€·[]tr(𝑉𝐴)(π‘Ÿ)(𝑉𝐡)(π‘Ÿ)βˆ—ξ€Έπ‘‘πœ†(π‘Ÿ),π΄βˆˆπ΅π‘ξ€·β„“2ξ€Έ,(3.17) with β€–π΅β€–π΅π‘ž(β„“2)≀‖𝐢(β‹…)‖𝐿𝑝(𝐷,π‘‘πœ†,β„“2)=β€–πΉβˆ˜π‘‰βˆ’1β€–β‰€β€–π‘‰βˆ’1‖‖𝐹‖. This proves the duality 𝐡𝑝(β„“2)βˆ—β‰ˆπ΅π‘ž(β„“2) for 1≀𝑝<∞.
It remains to prove the duality β„¬βˆ—0,𝑐(𝐷,β„“2)β‰ˆπ΅1(β„“2).
Let us assume that 𝐹 is a bounded linear functional on ℬ0,𝑐(𝐷,β„“2). Then we will prove that there is a matrix 𝐢 from 𝐡1(β„“2) such that ξ€œπΉ(𝐡)=10ξ€Ίtr𝑉𝐡(π‘Ÿ)(𝑉𝐢)βˆ—ξ€»(π‘Ÿ)π‘‘πœ†(π‘Ÿ),(3.18) for 𝐡 from a dense subset of ℬ0(𝐷,β„“2). By Lemma 1.2 it follows that π‘‰βˆΆβ„¬0(𝐷,β„“2)β†’π’ž0(𝐷,β„“2) is an isomorphic embedding. Thus 𝑋=𝑉(ℬ0,𝑐(𝐷,β„“2)) is a closed subspace in π’ž0(𝐷,𝐢∞) and 𝐹∘(𝑉)βˆ’1βˆΆπ‘‹β†’β„‚ is a bounded linear functional on 𝑋, where β„‚0(𝐷,𝐢∞) is the subset in π’ž0(𝐷,β„“2) whose elements are 𝐢∞-valued functions. By the Hahn-Banach theorem 𝐹∘(𝑉)βˆ’1 can be extended to a bounded linear functional on π’ž0(𝐷,𝐢∞).
Let Ξ¦βˆΆπ’ž0(𝐷,𝐢∞)β†’β„‚ denote this functional. It follows that π’ž0(𝐷,𝐢∞)=π’ž0ξβŠ—[0,1]πœ–πΆβˆž and, thus, Ξ¦ is a bilinear integral map, that is, there is a bounded Borel measure πœ‡ on [0,1]Γ—π‘ˆπΆ1, where π‘ˆπΆ1 is the unit ball of the space 𝐢1 with the topology 𝜎(𝐢1,𝐢∞), such that ξ€œΞ¦(π‘“βŠ—π΄)=[0,1]Γ—π‘ˆπΆ1𝑓(π‘Ÿ)trπ΄π΅βˆ—ξ€Έπ‘‘πœ‡(π‘Ÿ,𝐡)(3.19) for every π‘“βˆˆπ’ž0[0,1] and 𝐴∈𝐢∞.
Thus, for the matrix βˆ‘π‘›π‘˜=0π΄π‘˜βˆˆβ„¬0,𝑐(𝐷,β„“2), identified with the analytic matrix βˆ‘π‘›π‘˜=0π΄π‘˜π‘Ÿπ‘˜, we have that πΉξƒ©π‘›ξ“π‘˜=0π΄π‘˜ξƒͺ=πΉπ‘›ξ“π‘˜=0π‘Ÿπ‘˜π΄π‘˜ξƒͺ=ξ€ΊπΉβˆ˜(𝑉)βˆ’1ξ€»ξƒ¬π‘‰ξƒ©π‘›ξ“π‘˜=0π‘Ÿπ‘˜π΄π‘˜ξƒ©ξƒͺξƒ­=Ξ¦π‘›ξ“π‘˜=0(π‘˜+3)(π‘˜+2)2π‘Ÿπ‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΄π‘˜ξƒͺ=ξ€œ[0,1]Γ—π‘ˆπΆ1π‘›ξ“π‘˜=0trξ‚Έξ‚΅(π‘˜+3)(π‘˜+2)2π‘Ÿπ‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΄π‘˜ξ‚Άπ΅βˆ—ξ‚Ήπ‘‘πœ‡(π‘Ÿ,𝐡)def=ξ„”ξƒ©πœ‡(π‘Ÿ,𝐡),trπ‘›ξ“π‘˜=0(π‘˜+3)(π‘˜+2)2π‘Ÿπ‘˜π΄π‘˜ξƒͺπ΅βˆ—ξ€·1βˆ’π‘Ÿ2ξ€Έ2ξ„•.(3.20) On the other hand, we wish to have that ξ€œπΉ(𝐴)=10tr𝑉(𝐴)(𝑉(𝐢))βˆ—=ξ€œπ‘‘πœ†(𝑠)10trπ‘›ξ“π‘˜=0(π‘˜+3)(π‘˜+2)2π‘ π‘˜π΄π‘˜ξƒͺ(𝑉(𝐢))βˆ—=ξ€œ(2𝑠𝑑𝑠)10trπ‘›ξ“π‘˜=0𝑠2π‘˜(π‘˜+3)2(π‘˜+2)24ξ€·1βˆ’π‘ 2ξ€Έ2π΄π‘˜πΆβˆ—π‘˜ξƒͺ=(2𝑠𝑑𝑠)π‘›ξ“π‘˜=0trπ΄π‘˜ξ‚΅(π‘˜+3)(π‘˜+2)𝐢2(π‘˜+1)βˆ—π‘˜ξ‚Ά.(3.21) Therefore, letting 𝐴=𝑒𝑖,𝑖+π‘˜, denote the matrix having 1 as the single nonzero entry on the 𝑖th-row and the (𝑖+π‘˜)th-column, for 𝑖β‰₯1 and 𝑗β‰₯0, we have that πΆπ‘˜=ξ‚¬πœ‡(π‘Ÿ,𝐡),(π‘˜+1)π‘Ÿπ‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΅π‘˜ξ‚­,π‘˜=0,1,2,….(3.22) Then, it yields that ξ€œ10β€–β€–πΆξ…žξ…žβ€–β€–(𝑠)𝐢1=ξ€œ2𝑠𝑑𝑠10β€–β€–β€–β€–ξ€œ[0,1]Γ—π‘ˆπΆ1π‘›ξ“π‘˜=2(π‘˜+1)!𝑠(π‘˜βˆ’2)!π‘˜βˆ’2π‘Ÿπ‘˜ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΅π‘˜β€–β€–β€–β€–π‘‘πœ‡(π‘Ÿ,𝐡)𝐢1β‰€ξ€œ(2𝑠𝑑𝑠)[0,1]Γ—π‘ˆπΆ1βŽ‘βŽ’βŽ’βŽ£ξ€œ10β€–β€–β€–β€–π‘›ξ“π‘˜=2(π‘˜+1)!(π‘˜βˆ’2)!(π‘Ÿπ‘ )π‘˜βˆ’2π‘Ÿ2ξ€·1βˆ’π‘Ÿ2ξ€Έ2π΅π‘˜β€–β€–β€–β€–πΆ1⎀βŽ₯βŽ₯βŽ¦π‘‘||πœ‡||β‰€ξ€œ(2𝑠𝑑𝑠)(π‘Ÿ,𝐡)[0,1]Γ—π‘ˆπΆ1βŽ‘βŽ’βŽ’βŽ£ξ€œ10β€–β€–β€–β€–π‘›ξ“π‘˜=2(π‘˜+1)!(π‘˜βˆ’2)!(π‘Ÿπ‘ )π‘˜βˆ’2π‘Ÿ2ξ€·1βˆ’π‘Ÿ2ξ€Έ2π‘’π‘–π‘˜(β‹…)‖‖‖‖𝐿1(𝕋)‖𝐡‖𝐢1⎀βŽ₯βŽ₯βŽ¦π‘‘||πœ‡||β‰€ξ€œ(2𝑠𝑑𝑠)(π‘Ÿ,𝐡)[0,1]Γ—π‘ˆπΆ1ξƒ©ξ€œ10ξ€œ02πœ‹π‘Ÿ2ξ€·1βˆ’π‘Ÿ2ξ€Έ2||1βˆ’π‘Ÿπ‘ π‘’π‘–πœƒ||4π‘‘πœƒξƒͺ𝑑||πœ‡||βˆΌξ€œ2πœ‹(2𝑠𝑑𝑠)(π‘Ÿ,𝐡)[0,1]Γ—π‘ˆπΆ1π‘Ÿ2ξ€·1βˆ’π‘Ÿ2ξ€Έ21ξ€·1βˆ’π‘Ÿ2ξ€Έ2𝑑||πœ‡||(π‘Ÿ,𝐡)β‰€β€–πœ‡β€–<∞.(3.23) Consequently, 𝐢∈𝐡1(β„“2) and we get the relation (3.18) by using the fact that the set of all matrices βˆ‘π‘›π‘˜=0π΄π‘˜ is dense in ℬ0,𝑐(𝐷,β„“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 𝐡1(β„“2).

Theorem 3.3. One has(𝐡1(β„“2),𝐡1(β„“2))=𝐻11,∞,1(β„“2)def={𝐴∢supπ‘Ÿ<1βˆ‘(1βˆ’π‘Ÿ)β€–π‘˜β‰₯0π‘˜π΄π‘˜β€–π‘€(β„“2)<∞}.

Proof. By Lemma 3.1 we have that 𝑉(π΄βˆ—π΅)=𝑉(𝐴)βˆ—π΅ for all 𝐴∈𝐡1(β„“2) and for all matrices 𝐡 such that π΄βˆ—π΅βˆˆπ΅1(β„“2). Consequently (𝐡1(β„“2),𝐡1(β„“2))=(ℬ(β„“2),ℬ(β„“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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