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=(ajk)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:
A*B=(ajkbjk)j,k≥0.
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},
equipped with the natural norm
‖M‖=sup‖A‖X≤1‖M*A‖Y.
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
μ̂(n)=cn,forn=0,±1,±2,….
Moreover, one then has that
‖M‖=‖μ‖.

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(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=(aij) and an integer k we denote by Ak the matrix whose entries a′ij are given by
a′ij={aijifj-i=k,0otherwise.
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,
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-r2)‖A′(r)‖B(l2)+‖A0‖B(l2)<∞,
where B(ℓ2) is the usual operator norm of the matrix A on the sequence space ℓ2 andA′(r)=∑k=0∞Akkrk-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 ∫01∥A(r)∥Cpp2rdr<∞ and for all i,j∈ℕ we have that
[P(A(⋅))](r)(i,j)={2(j-i+1)rj-i∫01aij(s)⋅sj-i+1ds,ifi≤j,0,otherwise.

Now we consider a modified version of Bergman projection.

Let α>-1. Then[PαA(⋅)](r)={(α+1)Γ(j-i+2+α)(j-i)!Γ(α+2)rj-i(2∫01aij(s)sj-i+1(1-s2)αds)ifj≥i0ifj<i.

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∫01aij(s)sj-i(2sds)ifj-i≥00otherwise.

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 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.
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)=[∫01(1-r2)2p‖A′′(r)‖Cppdλ(r)]1/p<∞.

On Bp(ℓ2) we introduce the norm
‖A‖=‖A0‖C1+‖A‖Bp(l2).

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)=(∫01‖A(r)‖Cppdλ(r))1/p<∞.

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).
Then,

if c<0, then Ic,t(z) is bounded in z;

if c>0, then
Ic,t(z)~1(1-|z|2)c(|z|⟶1-);

if c=0, then
I0,c(z)~log11-|z|2(|z|⟶1-).

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 r→A′′(r) is continuous on [0,r0) for some 1>r0>0. Then the following assertions are equivalent:

A∈Bp(ℓ2);

(1-r2)2A′′(r)∈Lp(D,dλ,ℓ2);

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(⋅))=∑k=0∞(k+1)∫01[A(s)]ksk(2sds),
where A(·)∈Lp(D,ℓ2). Then
P((1-r2)2Akrk)(s)=2skAk(k+2)(k+3),
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=∑k=04Ak+A1,
where A1:=∑k=5∞Ak.

If (1-r2)2A′′(r)∈Lp(D,dλ,ℓ2), then we have that
Φ(r):=∑k=04(k+2)(k+3)2(1-r2)2Akrk+(1-r2)2(A1)′′(r)2!r2
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
∫0r0‖(A1)′′(s)‖Cpps2pds<∞.
Consequently Φ∈Lp(D,dλ,ℓ2).

Moreover A=PΦ since
∑k=5∞∫01k(k-1)Aksk-2(1-s2)2s2(k+1)sk+1ds=∑k=5∞(k-1)k(k+1)Ak∫01s2k-3(1-s2)2ds=∑k=5∞Ak.
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)2A′′(r)=(1-r2)2∫01[Φ(s)*6s2(1-rs)4]2sds.
Using Fubini’s theorem and Lemma 2.2 we obtain that
∫01(1-r2)2‖A′′(r)‖C12rdr(1-r2)2≤∫01[∫01‖Φ(s)‖C1∫02π6s2dθ|1-rseiθ|42sds]2rdr=∫016s2‖Φ(s)‖C1[∫0112π∫02πdθ|1-rseiθ|42rdr]2sds~∫01‖Φ(s)‖C112s3(1-s2)2ds≤6∫01‖Φ(s)‖C1dλ(s)<∞.

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))i,j=1∞ be the Toeplitz matrix with
tij(rs)=tj-i(rs)={s2(rs)j-i(j-i+3)(j-i+2)(j-i+1)ifj≥i0otherwise.
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)2‖A′′(r)‖Cp=(1-r2)2‖∫01[ϕ(s)*6s2(1-rs)4](2sds)‖Cp≤(1-r2)2∫01‖ϕ(s)‖Cp‖6s2(1-rseiθ)4‖L1(T)(2sds)=(1-r2)2∫01‖ϕ(s)‖Cp(1-s2)2‖6s2(1-rseiθ)4‖L1(T)dλ(s):=Sϕ(r).
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)2A′′(r)∈Lp(D,dλ,l2)
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)={(j-i+3)(j-i+2)(j-i+1)2rj-i(1-r2)2∫01aij(s)sj-i(2sds)ifj-i≥0,0otherwise.

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
P(B(⋅))(r)(i,j)={2(j-i+1)rj-i∫01bij(s)sj-i+1dsifj-i≥00otherwise,
it is easy to see that
PV=P,VP=V
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)=∫01‖[V(A(⋅))]‖C1dλ(r)=∫01‖∑k=0∞(k+3)(k+2)(k+1)2rk(1-r2)2∫01Ak(s)sk(2sds)‖C1dλ(r)≤∬01‖∑k=0∞(k+3)(k+2)(k+1)2rk(1-r2)2Ak(s)sk+1‖C1(2ds)dλ(r)=∫01[∫01‖∑k=0∞(k+3)(k+2)(k+1)2rkAk(s)sk(1-s2)2‖C1(2rdr)]dλ(s)=∬01‖A(s)*C(rs)‖C1(2rdr)dλ(s),
where C(rs)=(cij(rs))i,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.
Since the Toeplitz matrix C(rs) is a Schur multiplier with
‖C(rs)‖M(l2)=‖6s2(1-s2)2(1-rseiθ)4‖L1(T),
then, according to Lemma 2.2, it follows that
∬01‖A(s)*C(rs)‖C1(2rdr)dλ(s)≤∫01‖A(s)‖C1∫01‖C(rs)‖M(l2)(2rdr)dλ(s)~∫01‖A(s)‖C1dλ(s).
Consequently V is bounded on L1(D,dλ,ℓ2). For 1<p<∞ we have that
‖VA(⋅)(r)‖Cp≤∫01‖∑k=0∞(k+3)(k+2)(k+1)2rksk(1-r2)2(1-s2)2Ak(s)‖Cpdλ(s)=∫01‖A(s)*T(rs)‖Cpdλ(s),
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.T(rs)is a Schur multiplier, therefore
∫01‖A(s)*T(rs)‖Cpdλ(s)≤∫01‖A(s)‖Cp(1-r2)2(1-s2)2‖6(1-rseiθ)4‖L1(T):=SA(r).
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)
for all A=PB(·). Taking the infimum over B, we get that
‖V(A)‖Lp(D,dλ,l2)≤C‖A‖Bp(l2).
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),
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〉=∫01tr(V(A)[V(B)]*)dλ(r)
One has the following dualities:

Bp(ℓ2)*≈Bq(ℓ2) if 1<p<∞ and 1/p+1/q=1;

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 Bq(ℓ2)⊂Bp(ℓ2)* for 1≤p<∞ and B1(ℓ2)⊂ℬ0,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(·)∈Lq(D,dλ,ℓ2) such that ∥C(·)∥Lq(D,dλ,ℓ2)=∥F∘V-1∥ and
(F∘V-1)(B)=∫01tr(B(r))[C(r)]*dλ(r),B(⋅)∈Lp(D,dλ,l2).
In particular, if B(·)=V(A) with A∈Bp(ℓ2), then
F(A)=∫01tr((VA)(r))[C(r)]*dλ(r).
Let B=P(C). Then B∈Bq(ℓ2) and it is easy to check that
F(A)=∫01tr((VA)(r)[(VB)(r)]*)dλ(r),A∈Bp(l2),
with ∥B∥Bq(ℓ2)≤∥C(·)∥Lp(D,dλ,ℓ2)=∥F∘V-1∥≤∥V-1∥∥F∥. This proves the duality Bp(ℓ2)*≈Bq(ℓ2) for 1≤p<∞.

It remains to prove the duality ℬ0,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)=∫01tr[VB(r)(VC)*(r)]dλ(r),
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(ℬ0,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]×UC1, where UC1 is the unit ball of the space C1 with the topology σ(C1,C∞), such that
Φ(f⊗A)=∫[0,1]×UC1f(r)tr(AB*)dμ(r,B)
for every f∈𝒞0[0,1] and A∈C∞.

Thus, for the matrix ∑k=0nAk∈ℬ0,c(D,ℓ2), identified with the analytic matrix ∑k=0nAkrk, we have that
F(∑k=0nAk)=F(∑k=0nrkAk)=[F∘(V)-1][V(∑k=0nrkAk)]=Φ(∑k=0n(k+3)(k+2)2rk(1-r2)2Ak)=∫[0,1]×UC1∑k=0ntr[((k+3)(k+2)2rk(1-r2)2Ak)B*]dμ(r,B)=def〈μ(r,B),tr(∑k=0n(k+3)(k+2)2rkAk)B*(1-r2)2〉.
On the other hand, we wish to have that
F(A)=∫01trV(A)(V(C))*dλ(s)=∫01tr(∑k=0n(k+3)(k+2)2skAk)(V(C))*(2sds)=∫01tr(∑k=0ns2k(k+3)2(k+2)24(1-s2)2AkCk*)(2sds)=∑k=0ntrAk((k+3)(k+2)2(k+1)Ck*).
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,….
Then, it yields that
∫01‖C′′(s)‖C12sds=∫01‖∫[0,1]×UC1∑k=2n(k+1)!(k-2)!sk-2rk(1-r2)2Bkdμ(r,B)‖C1(2sds)≤∫[0,1]×UC1[∫01‖∑k=2n(k+1)!(k-2)!(rs)k-2r2(1-r2)2Bk‖C1(2sds)]d|μ|(r,B)≤∫[0,1]×UC1[∫01‖∑k=2n(k+1)!(k-2)!(rs)k-2r2(1-r2)2eik(⋅)‖L1(T)‖B‖C1(2sds)]d|μ|(r,B)≤∫[0,1]×UC1(∫01∫02πr2(1-r2)2|1-rseiθ|4dθ2π(2sds))d|μ|(r,B)~∫[0,1]×UC1r2(1-r2)21(1-r2)2d|μ|(r,B)≤‖μ‖<∞.
Consequently, C∈B1(ℓ2) and we get the relation (3.18) by using the fact that the set of all matrices ∑k=0nAk 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.

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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