A Class of Schur Multipliers on Some Quasi-Banach Spaces of Infinite Matrices

Abstract

We characterize the Schur multipliers of scalar type acting on scattered classes of infinite matrices.

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In [1], Schur introduced a new product between two matrices 𝐴=(𝑎_𝑗𝑘) and 𝐵=(𝑏_𝑗𝑘) of the same size, finite or infinite. This product, known in the literature as the Schur product or Hadamard product, is defined to be the matrix of elementwise products𝐴∗𝐵=𝑎_𝑗𝑘𝑏_𝑗𝑘.(1)

This concept was used in different areas of analysis as complex function theory, Banach spaces, operator theory, and multivariate analysis.

Bennett studied in [2] the behaviour, under Schur multiplication, of the norm ‖⋅‖_𝑝,𝑞,1≤𝑝,𝑞≤∞,‖𝐴‖_𝑝⋅𝑞=sup_{‖𝑥‖𝑝≤1}⎛⎜⎜⎝_𝑗|||||_𝑘𝑎_𝑗𝑘𝑥_𝑘|||||^𝑞⎞⎟⎟⎠^1/𝑞.(2)

In particular, he was interested in characterizing the (𝑝,𝑞)-multipliers: the matrices 𝑀 for which 𝑀∗𝐴 maps ℓ_𝑝 into ℓ_𝑞 whenever 𝐴 does.

In his paper it is proved a theorem about Schur multipliers which are Toeplitz matrices, that is about the matrices of the form𝐴=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎⁰𝑎₋₁𝑎₁𝑎₂𝑎₃⋯𝑎₋₂𝑎₀𝑎₁𝑎₂⋯𝑎₋₃𝑎₋₁𝑎₀𝑎₁⋯⋮𝑎⋱₋₂𝑎⋱₋₁⋱𝑎₀⋱⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3) where (𝑎_𝑗)^{∞𝑗=−∞} is a sequence of complex numbers.

Theorem 8.1 in [2] reads as follows.

Theorem B. A Toeplitz matrix 𝐴 is a Schur multiplier if and only if 𝜇=∑^{∞𝑗=−∞}𝑎_𝑗𝑒^𝑖𝑗𝑡 is a bounded Borel measure on [0,2𝜋).

This fact leads naturally to the idea of identifying the Schur multipliers with the noncommutative bounded Borel measures, see, for example, [3].

We denote by 𝑀(ℓ₂) the space of all (2,2) Schur multipliers from 𝐵(ℓ₂) into 𝐵(ℓ₂), where 𝐵(ℓ₂) is, as usual, the Banach space of linear and bounded operators on ℓ₂ with the usual operator norm.

The space 𝑀(ℓ₂) endowed with norm ‖𝐴‖_𝑀(ℓ2)=sup_{‖𝐵‖2𝐵(ℓ)≤1}‖𝐴∗𝐵‖_𝐵(ℓ2) becomes a Banach space.

Since we work with different quasi-Banach spaces of matrices 𝑋,𝑌 we use the notation (𝑋,𝑌) for the space of all Schur multipliers from 𝑋 into 𝑌 equipped with the quasi-norm‖𝐴‖_(𝑋,𝑌)=sup_{‖𝐵‖𝑌≤1}‖𝐴∗𝐵‖_𝑋.(4)

In this way (𝑋,𝑌) becomes a quasi-Banach.

In [2] Bennett raised the problem of characterizing the Hankel matrices which are Schur multipliers.

We recall that a matrix 𝐴 is called a Hankel matrix if it is defined by a sequence (𝑎_𝑗)^∞𝑗=1 of complex numbers in the following way:𝐴=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎⁰𝑎₁𝑎₁𝑎₂𝑎₃⋯𝑎₂𝑎₂𝑎₃𝑎₄⋯𝑎₃𝑎₃𝑎₄𝑎₅⋯⋮𝑎⋮₄𝑎⋮₅𝑎⋮₆⋯⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(5)

Pisier in [4] solved the above problem. He proved the following theorem.

Theorem P. A Hankel matrix is a Schur multiplier if and only if the Fourier multiplier ∑^∞𝑛=0𝑥_𝑛𝑒^int→∑^∞𝑛=0𝑎_𝑛𝑥_𝑛𝑒^int maps boundedly 𝐻¹(𝑆₁) into itself.

Here 𝐻¹(𝑆₁) is the Hardy space of the Schatten class 𝑆₁-valued analytic functions, endowed with the norm ‖𝑓‖_𝐻¹_(𝑆1)=(1/2𝜋)∫^02𝜋‖∑^∞𝑛=0𝐴_𝑛𝑒^int‖_𝑆1𝑑𝑡<∞. For the definition of the Schatten classes 𝑆_𝑝, see, for example, [5].

In [5], Aleksandrov and Peller characterized the Toeplitz matrices which are Schur multipliers for 𝑆_𝑝, 0<𝑝<1. They proved the following theorem.

Theorem AP. Let 0<𝑝<1. A Toeplitz matrix 𝑇 given by the complex sequence (𝑡_𝑗)^{∞𝑗=−∞} belongs to (𝑆_𝑝,𝑆_𝑝) if and only if there exists a measure 𝜇∈𝑀_𝑝 with the Fourier coefficients 𝜇(𝑗)=𝑡_𝑗. Moreover, in this case ‖𝑇‖_{(𝑆𝑝,𝑆𝑝)}=‖𝜇‖_𝑀𝑝,(6) where 𝑀_𝑝={𝜇∶𝕋→ℂ∣𝜇=∑_𝑗𝛼_𝑗𝛿_𝑡𝑗,𝑡_𝑗∈𝕋,𝑡_𝑗𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡𝑝𝑜𝑖𝑛𝑡𝑠},‖𝜇‖_𝑀𝑝=(∑_𝑗|𝛼_𝑗|^𝑝)^1/𝑝<∞, and 𝛿_𝑡 is the Dirac measure concentrated at the point 𝑡∈𝕋.

The above-mentioned papers [4, 5] show that a complete description of general Schur multipliers, at least, either for 𝐵(ℓ₂) or 𝑆_𝑝,0<𝑝≤1, is a difficult target. In this way it is natural to consider and study other classes of Schur multipliers than those which are Toeplitz matrices. In [6], the following notation, more apropriate for our aims, for the entries of a matrix 𝐵 was introduced. Namely, we put𝑏^𝑙𝑘=𝑏^{𝑙,𝑙+𝑘}𝑏,𝑘≥0,𝑙=1,2,…,_{𝑙−𝑘,𝑙},𝑘<0,𝑙=1,2,…,(7) and write 𝐵=(𝑏^𝑙𝑘)_{𝑙≥1,𝑘∈ℤ}.

Let 𝐵^(𝑙)=(𝑏^𝑚𝑘)_{𝑘∈ℤ,𝑚≥1}, where 𝑙=1,2,3,…, be the matrix given by𝑏^𝑚𝑘=𝑏^𝑙𝑘0,,iiff𝑚𝑚≠=𝑙𝑙.,(8)

We call the matrix 𝐵^(𝑙), the 𝑙th corner matrix associated to 𝐵.

Now, we associate to each matrix 𝐵^𝑙 a periodical distribution on 𝕋, denoted by 𝑓_𝑙, such that 𝑏^𝑙𝑘=𝑓_𝑙(𝑘), and we identify the matrix 𝐵=(𝐵^(𝑙))_𝑙∈ℕ^∗ with the sequence of associated distributions (𝑓_𝑙)_𝑙∈ℕ^∗.

Then for the sequence 𝛼=(𝛼¹,𝛼²,…) and the matrix 𝐵=(𝑓_𝑙)_𝑙∈ℕ^∗, we denote by 𝛼⊙𝐵 the matrix given by (𝛼^𝑙𝑓_𝑙)_{𝑙∈ℕ∗}.

In particular, if 𝐵 is a Toeplitz matrix (𝐵∈𝒯) and if 𝛼 is the constant sequence then 𝛼⊙𝐵 coincides with the matrix 𝛼𝐵.

Hence, if [𝛼] is the matrix[𝛼]=⎛⎜⎜⎜⎜⎜⎜⎝𝛼¹𝛼₁𝛼¹𝛼¹⋯𝛼₁𝛼²𝛼²⋱⋮⋱𝛼²⋱𝛼³⋱⋱⎞⎟⎟⎟⎟⎟⎟⎠(9) it is clear that 𝛼⊙𝐵=[𝛼]∗𝐵.

We define 𝑚𝑠 to be the space of all sequences 𝛼 such that 𝛼⊙𝐵∈𝐵(ℓ₂) for all 𝐵∈𝐵(ℓ₂), or equivalently [𝛼]∈𝑀(ℓ₂).

On 𝑚𝑠 we consider the norm ‖𝛼‖_𝑚𝑠=‖[𝛼]‖_𝑀(ℓ2). Then 𝑚𝑠 is a unital commutative Banach algebra with respect to the usual multiplication of sequences. As it was observed in [6], the multiplication of a function with a scalar corresponds to the multiplication ⊙ of a sequence and an infinite matrix.

We call the matrices [𝛼]  scalar matrices. In this context, in [6] a theorem of Haar’s type for infinite matrices was proved. The product ⊙ appeared also in [7] in other contexts.

An important role in applications is played by the upper triangular projection applied to the matrix [𝛼]. For an infinite matrix 𝐴=(𝑎_𝑖𝑗)_{𝑖≥1,𝑗≥1,} the upper triangular projection is𝑃_𝑇(𝐴)=𝑎^𝑖,𝑗0,,iofth𝑖e≤rw𝑗i,se.(10)

A sequence 𝑏=(𝑏_𝑛)_𝑛≥1 belongs to 𝑝𝑚𝑠 if and only if𝐵={𝑏}=𝑃_𝑇([𝑏])∈𝑀ℓ₂.(11)

The space 𝑝𝑚𝑠 endowed with the norm ‖𝑏‖=‖{𝑏}‖_𝑀(ℓ2) becomes a Banach algebra with respect to the usual product of sequences.

In [6] there were given sufficient and necessary conditions in order for matrices of the form [𝛼] or {𝛼}, that is,{𝛼}=⎛⎜⎜⎜⎜⎜⎜⎝𝛼¹0𝛼𝛼¹₂𝛼¹⋯00𝛼𝛼²₃⋱⋮⋱⋱⋱⋱⎞⎟⎟⎟⎟⎟⎟⎠,(12) to be Schur multipliers. The following result was proved.

Theorem BLP1. Let 𝑏=(𝑏_𝑛)_𝑛≥1 be a complex sequence. (1)If (𝑖_𝑛)_𝑛≥1 is a strictly increasing sequence of natural numbers with 𝑖₁=0, and 𝑧_𝑖𝑛=max_{𝑖𝑛<𝑘≤𝑖𝑛+1}|𝑏_𝑘|, then there is a constant 𝑅>0 such that‖{𝑏}‖_𝑀(ℓ2)≤𝑅inf_{(𝑖𝑛)𝑛≥1}‖‖𝑧^𝑖_𝑛_𝑛≥1‖‖₂+‖‖𝑧_𝑖𝑛log𝑖_𝑛+1−𝑖_𝑛_𝑛‖‖_∞.(13)(2)If 𝑏∈𝑝𝑚𝑠 thensup_{𝑛≥1;𝑝≥1}(log𝑛)²𝑛^𝑛+𝑝_𝑘=𝑝||𝑏_𝑘||²<∞.(14)(3)If (|𝑏_𝑘|)_𝑘≥1 is a decreasing sequence, then 𝑏∈𝑝𝑚𝑠 if and only if |𝑏_𝑘|=𝒪(1/log𝑘).

As an immediate consequence we have the following.

Corollary 1. (1) One has ℓ₂⊂𝑚𝑠⊂ℓ_∞.
(2) One has {(𝑏_𝑛)_𝑛≥1∣|𝑏_𝑛|=𝒪(1/log𝑛)}⊂𝑚𝑠.

A set of sufficient conditions in order for a matrix of the type [𝛼] to be a Schur multiplier is given in [6], namely, the following theorem was proved.

Theorem BLP2. Let 𝑏=(𝑏_𝑛)_𝑛≥1 a complex sequence. Then, (1)if sup_𝑛≥1∑^𝑛𝑗=1|𝑏_𝑗−𝑏_𝑛|²<∞, then 𝑏∈𝑚𝑠;(2)if ‖𝑏‖_𝐵𝑉ℕ=|𝑏₁|+∑^∞𝑛=1|𝑏_𝑛+1−𝑏_𝑛|<∞ then 𝑏∈𝑚𝑠.

It is well known that 𝑀(ℓ₂) coincides with (𝑆₁,𝑆₁), the space of all Schur multipliers from 𝑆₁ into 𝑆₁, see, for example, [4]. Using this fact we give a simpler proof of the first statement of Corollary 1.

Theorem 2. Let 𝑐=(𝑐_𝑛)_𝑛≥1∈ℓ₂. Then 𝑐∈𝑝𝑚𝑠.

Proof. By using the Schmidt decomposition of a matrix 𝐴, it is enough to show that 𝐴∗{𝑐}∈𝑆₁ for a matrix 𝐴 of rank 1. Let 𝐴=𝛼⊗𝛽 with 𝛼=(𝛼_𝑛)_𝑛≥1∈ℓ₂ and 𝛽=(𝛽_𝑛)_𝑛≥1∈ℓ₂.
We have 𝐴∗{𝑐}=⎛⎜⎜⎜⎜⎜⎜⎝𝛼¹𝛼₂𝛽₁𝛼₁𝛽₂𝛼₁𝛽₃⋯𝛼₃𝛽₁𝛼₂𝛽₂𝛼₂𝛽₃⋯⋮𝛽₁𝛼⋮₃𝛽₂𝛼⋮₃𝛽₃⋯⎞⎟⎟⎟⎟⎟⎟⎠∗⎛⎜⎜⎜⎜⎜⎜⎝𝑐¹0𝑐𝑐₁₂𝑐₁𝑐₁⋯00𝑐𝑐₂₃𝑐₂⋱=⎛⎜⎜⎜⎜⎜⎜⎝𝑐⋮⋱⋱⋱𝑐₃⋱₁⋱⎞⎟⎟⎟⎟⎟⎟⎠𝛼0₁𝛽₁𝑐𝑐₁₂𝛼₁𝛽₂𝑐₁𝛼₁𝛽₃⋯0𝛼0₂𝛽₂𝑐𝑐₂₃𝛼₂𝛽₃⋯⋮⋮𝛼⋮₃𝛽₃⋯⎞⎟⎟⎟⎟⎟⎟⎠.(15)
By the definition of 𝑆₁ and Cauchy-Schwartz inequality we get ‖𝐴∗{𝑐}‖_𝑆1≤^∞_𝑗=1‖‖𝛼_𝑗𝑐_𝑗𝛽_𝑗+𝑘^∞𝑘=0‖‖_ℓ2=^∞_𝑗=1^∞_𝑘=0||𝛼_𝑗𝑐_𝑗||²||𝛽_𝑗+𝑘||²^1/2≤^∞_𝑗=1||𝛼_𝑗𝑐_𝑗||‖‖𝛽_𝑗‖‖_ℓ2≤‖𝛼‖_ℓ2‖𝛽‖_ℓ2‖𝑐‖_ℓ2=‖𝑐‖_ℓ2‖𝐴‖_𝑆1,(16) that is, ‖𝐴‖_𝑀(ℓ2)≤‖𝑐‖_ℓ2 and the proof is complete.

We characterize now the upper triangular scalar matrices which are Schur multipliers, from the Hardy space 𝐻², respectively, from the Schatten class 𝑆₂ into 𝐵(ℓ₂).

Theorem 3. (1) Let 𝐻² be the Hardy space of Toeplitz matrices generated by the classical Hardy space of functions. Then an upper triangular matrix 𝐴={𝛼} belongs to (𝐻²,𝐵(ℓ₂)) if and only if 𝛼∈ℓ₂. Moreover, one has equality of the norms.
(2) Let 𝑇₂ be the space of all upper triangular Hilbert-Schmidt matrices. Then {𝛼}∈(𝑇₂,𝐵(ℓ₂)) if and only if 𝛼∈ℓ_∞.

Proof. (1) We use the following identity proved in [6]: ‖𝐵‖_𝐵(ℓ2)=sup_{‖ℎ‖2≤1;ℎ∈𝐻20[]0,1}⎛⎜⎜⎝^∞_𝑘=1|||||_10∞_𝑗=𝑘𝑏_𝑘𝑗𝑒^{2𝜋𝑖𝑗𝑡}ℎ(−𝑡)𝑑𝑡|||||²⎞⎟⎟⎠^1/2,(17) where 𝐵 is an upper triangular matrix 𝐵=(𝑏_𝑘𝑗).
Then, if 𝑓(𝑡)=∑^∞𝑘=0𝑐_𝑘𝑒^{2𝜋𝑖𝑘𝑡}∈𝐻²,𝑡∈[0,1], 𝐹 is the Toeplitz matrix associated to 𝑓 (i.e., 𝐹 is given by (𝑐_𝑘)^∞𝑘≥0), and 𝛼∈ℓ₂, we have 𝐹∗{𝛼}=⎛⎜⎜⎜⎜⎜⎜⎝𝛼⁰0𝑐₀𝛼𝛼₀₁𝑐₁𝛼₀𝑐₂𝛼₀𝑐₃⋯00𝑐₀𝛼𝛼₁₂𝑐₁𝛼₁𝑐₂⋱‖𝐹∗{𝛼}‖_𝐵(ℓ2)⋮⋱⋱𝑐₀𝛼⋱₂𝑐₁⋱⋱⎞⎟⎟⎟⎟⎟⎟⎠=sup_{‖ℎ‖2≤1}⎛⎜⎜⎝^∞_𝑘=1|||||₁₀𝛼_𝑘−1^∞_𝑗=0𝑐_𝑗𝑒^{2𝜋𝑖𝑗𝑡}ℎ(−𝑡)𝑑𝑡|||||²⎞⎟⎟⎠^1/2=sup_{‖ℎ‖2≤1}^∞_𝑘=1||𝛼_𝑘−1||²^1/2|||||₁₀^∞_𝑗=0𝑐_𝑗𝑒^{2𝜋𝑖𝑗𝑡}ℎ(−𝑡)𝑑𝑡|||||=‖𝛼‖^ℓ2‖𝑓‖_𝐻².(18)
Hence ‖{𝛼}‖_(𝐻²_,𝐵(ℓ2))=‖𝛼‖_ℓ2, and this completes the proof.
(2) Let 𝛼∈ℓ_∞ and 𝐶=⎛⎜⎜⎜⎜⎜⎜⎝𝑐¹¹0𝑐𝑐₁₂₂₂𝑐₁₃⋯00𝑐𝑐₂₃₃₃⋯⋮⋮⋮⋯⎞⎟⎟⎟⎟⎟⎟⎠∈𝑇².(19) Using formula (17) and Cauchy-Schwartz inequality we get ‖{𝛼}∗𝐶‖_𝐵(ℓ2)≤⎛⎜⎜⎝^∞_𝑘=1||𝛼_𝑘−1||²sup_{‖ℎ‖2≤1}|||||₁₀_𝑗≥𝑘𝑐_𝑘𝑗𝑒^{2𝜋𝑖𝑗𝑡}ℎ(−𝑡)|||||²⎞⎟⎟⎠^1/2≤^∞_𝑘=1||𝛼_𝑘−1||²_𝑗≥𝑘||𝑐_𝑘𝑗||²^1/2≤sup_𝑘≥0||𝛼_𝑘||^∞_𝑘=1_𝑗≥𝑘||𝑐_𝑘𝑗||²^1/2=‖𝛼‖_ℓ∞‖𝐶‖_𝑇2.(20)
Hence {𝛼}∈(𝑇₂,𝐵(ℓ₂)), and this proves the first part of the theorem.
Conversely, let {𝛼}∈(𝑇₂,𝐵(ℓ₂)), that is, 𝐶∗{𝛼}∈𝐵(ℓ₂) for all 𝐶∈𝑇₂ and take 𝐶=𝐶₀, that is, the matrix 𝐶 is reduced to main diagonal. It is clear that the sequence of entries of this diagonal belongs to ℓ₂. Consequently the sequence (𝛼_𝑘−1𝑐_𝑘𝑘)_𝑘≥0 belongs to ℓ_∞ for every sequence (𝑐_𝑘𝑘)^∞𝑘≥1∈ℓ₂. Hence (𝛼_𝑘)^∞𝑘=0∈ℓ_∞, and the proof is complete.

Next we use the important results of Bennett proved in [8], in order to characterize the Schur multipliers of scalar type for some spaces of lower triangular infinite matrices contained in the Schatten classes 𝑆_𝑝,0<𝑝<∞. We denote these spaces by ℒ𝒯𝑆_𝑝.

Next we get a general description of upper triangular Schur multipliers of scalar type for different quasi-Banach spaces.

In order to state the following result we need to recall some definitions (see [9]).

Let 𝐟 be the space of all sequences with a finite number of nonzero elements. A norm Φ on 𝐟 is called symmetric if Φ(𝑎)=Φ(𝑎^∗), for all 𝑎∈𝐟, that is, if Φ is invariant to permutations and to applications 𝑎_𝑛→𝑒^𝑖𝜃𝑛𝑎_𝑛, where 𝜃_𝑛 is a sequence of real numbers. Here 𝑎^∗=(𝑎^∗𝑛)^∞𝑛=1 is the decreasing rearrangement of the sequence (𝑎_𝑛) which converges to 0.

We say that the sequence (𝑎_𝑛)_𝑛 belongs to the space 𝑠_Φ, if and only if lim_𝑛→∞Φ(𝑎₁,…,𝑎_𝑛,0,0,…)=Φ(𝑎) exists.

We denote by 𝑆_Φ the space of all compact operators 𝐴 on ℓ₂ with the sequence of their singular numbers (𝜇_𝑛(𝐴)) belonging to 𝑠_Φ. For 𝐴∈𝑆_Φ we put Φ(𝐴)=Φ((𝜇_𝑛(𝐴))_𝑛).

Then the following noncommutative Hölder type inequality proved in [9] holds.

Theorem AH. Let Φ₁,Φ₂,Φ₃ be symmetric norms such that if 𝑎∈𝑠_Φ2,𝑏∈𝑠_Φ3 then 𝑎𝑏∈𝑠_Φ1 and Φ₁(𝑎𝑏)≤Φ₂(𝑎)Φ₃(𝑏).(21) If 𝐴∈𝑆_Φ2,𝐵∈𝑆_Φ3, then 𝐴𝐵∈𝑆_Φ1 and Φ₁(𝐴𝐵)≤Φ₂(𝐴)Φ₃(𝐵).

Using this inequality we can state the following interesting result.

Theorem 4. Let 𝑠_Φ1=𝑠_Φ2𝑠_Φ3 (i.e., for each 𝛼∈𝑠_Φ1, there exist 𝛽∈𝑠_Φ2,𝛾∈𝑠_Φ3 such that 𝛼=𝛽𝛾, and Φ₁(𝛼)≈inf_{𝛼=𝛽𝛾}Φ₂(𝛽)Φ₃(𝛾)). Then a scalar matrix [𝛼]∈(ℒ𝒯𝑆_Φ2,ℒ𝒯𝑆_Φ1) if and only if 𝛼∈𝑠_Φ3.

Proof. Let first 𝐴∈ℒ𝒯𝑆_Φ1 and 𝛼∈𝑠_Φ3. Then it is clear that 𝐴∗[𝛼]=𝐴⋅𝐷_𝛼,(22) where 𝐷_𝛼=⎛⎜⎜⎜⎜⎜⎜⎝𝛼¹0𝛼0₂0⋯00𝛼0₃⋱⋮⋱⋱⋱⋱⎞⎟⎟⎟⎟⎟⎟⎠.(23)
By Theorem AH it follows that ‖𝐴∗[𝛼]‖_𝑆Φ1≤‖‖𝐴⋅𝐷_𝛼‖‖_𝑆Φ1≤‖𝐴‖_𝑆Φ2‖‖𝐷_𝛼‖‖_𝑆Φ3≤‖𝐴‖_𝑆Φ2‖𝛼‖_𝑠Φ3.(24) Hence [𝛼]∈(ℒ𝒯𝑆_Φ2,ℒ𝒯𝑆_Φ1), and this completes the first part of the proof.
For the reverse implication, take 𝐴 to be the main diagonal with the entries (𝑎_𝑗𝑗)^∞𝑗=1∈𝑠_Φ2 and [𝛼]∈(ℒ𝒯𝑆_Φ2,ℒ𝒯𝑆_Φ1).
Then 𝐴∗[𝛼]=𝐴⋅𝐷_𝛼=⎛⎜⎜⎜⎜⎜⎜⎝𝑎¹¹0𝛼₁𝑎0₂₂0⋯00𝛼₂𝑎0₃₃⋱⋮⋱⋱𝛼₃⋱⋱⎞⎟⎟⎟⎟⎟⎟⎠∈𝑆^Φ1(25) and we get that (𝑎_𝑖𝑖𝛼_𝑖)^∞𝑖=1∈𝑠_Φ1 for all sequences (𝑎_𝑗𝑗)^∞𝑗=1∈𝑠_Φ2. Since 𝑠_Φ1=𝑠_Φ2𝑠_Φ3, it follows that 𝛼∈𝑠_Φ3, and this completes the proof of the theorem.

Let 𝑤=(𝑤_𝑛) be a positive decreasing sequence of numbers. Of course the Lorentz space of sequences ℓ_𝑝,𝑤, 0<𝑝≤∞, is a space of the previous type 𝑠_Φ, see, for example, [9]. By the well-known fact that ℓ_𝑝,𝑤⋅ℓ_𝑞,𝑤=ℓ_𝑟,𝑤, for 1/𝑝+1/𝑞=1/𝑟;0<𝑝,𝑞,𝑟<∞ we get the following result.

Corollary 2. (1) Let 1/𝑝+1/𝑞=1/𝑟,0<𝑝,𝑞,𝑟<∞. Then [𝛼]∈(𝑆_𝑝,𝑆_𝑟) if and only if 𝛼∈ℓ_𝑞.
(2) Let 𝑤_𝑛 be a decreasing positive sequence, and let 0<𝑝, 𝑞<∞, be such that 1/𝑝+1/𝑞=1. Then [𝛼]∈(𝑆_𝑝,𝑤,𝑆_1,𝑤) if and only if 𝛼∈ℓ_𝑞,𝑤, where ℓ_𝑝,𝑤 is the weighted Lorentz space of sequences.

We call the Bergman-Schatten space of order 𝑝, 0<𝑝<∞, and we denote by 𝐿^𝑝𝑎(ℓ₂) the space of all upper triangular matrices 𝐴 such that ‖𝐴‖_𝐿^𝑝𝑎_(ℓ2)=(∫¹⁰‖∑^∞𝑘=0𝐴_𝑘𝑟^𝑘‖^𝑝𝑆_𝑝2𝑟𝑑𝑟)^1/𝑝<∞. See, for example, [10] for further notations and details.

By Hölder’s inequality we get the following result.

Theorem 6. Let 1≤𝑝<∞. Then [𝛼]∈(𝐿^𝑝𝑎(ℓ₂),𝐿^1𝑎(ℓ₂)) if and only if 𝛼∈ℓ_𝑞, where 1/𝑝+1/𝑞=1.

Proof. Let 𝐴∈𝐿^𝑝𝑎(ℓ₂) and 𝛼∈ℓ_𝑞. We clearly have that 𝐴∗[𝛼]=𝐷_𝛼⋅𝐴. By Theorem AH we get ‖𝐴∗[𝛼]‖_𝐿^1𝑎_(ℓ2)=≤¹⁰‖‖𝐷_𝛼⋅𝐴(𝑟)‖‖_𝑆12𝑟𝑑𝑟₁₀‖𝐴(𝑟)‖^𝑝𝑆_𝑝2𝑟𝑑𝑟^1/𝑝‖𝛼‖_ℓ𝑞=‖𝐴‖_𝐿^𝑝𝑎_(ℓ2)‖𝛼‖_ℓ𝑞,(26) that is, [𝛼]∈(𝐿^𝑝𝑎(ℓ₂),𝐿^1𝑎(ℓ₂)), and this completes the first part of the proof.
Conversely, let [𝛼]∈(𝐿^𝑝𝑎(ℓ₂),𝐿^1𝑎(ℓ₂)). By taking 𝐴=𝐴₀=(𝑎_𝑗𝑗)^∞𝑗=1∈𝐿^𝑝𝑎(ℓ₂), that is, for (𝑎_𝑗𝑗)_𝑗∈ℓ_𝑝, we get 𝐴∗[𝛼]=⎛⎜⎜⎜⎜⎝𝛼¹0𝑎₁₁𝛼₂0⋯⋮⋱𝑎₂₂⋱⋱⎞⎟⎟⎟⎟⎠∈𝐿^1𝑎ℓ₂,(27) or, equivalently, (𝛼_𝑗𝑎_𝑗𝑗)_𝑗∈ℓ₁. Hence by Hölder’s inequality it follows that (𝛼_𝑗)_𝑗∈ℓ_𝑞, and the proof is complete.

Using the results of Bennett, proved in [8] we can also describe the Schur multipliers of scalar type also for others quasi-Banach spaces of matrices. The spaces of sequences 𝑑(𝐚,𝑝), 𝑔(𝐚,𝑝), and ces(𝑝) were defined in [8].

We denote now by 𝑑^𝑞𝑀(𝐚,𝑝),𝑔^𝑞𝑀(𝐚,𝑝),ces^𝑞𝑀(𝑝), and ℓ^𝑞𝑀(𝑝) the spaces of upper triangular infinite matrices 𝐴=∑^∞𝑘=0𝐴_𝑘, with all the sequences on the diagonals belonging to 𝑑(𝐚,𝑝) (resp., 𝑔(𝐚,𝑝),ces(𝑝), ℓ_𝑝,) and such that ‖𝐴‖=(∑_𝑘‖𝐴_𝑘‖^{𝑞𝑑(𝐚,𝑝)})^1/𝑞<∞ (resp., ‖𝐴‖=(∑_𝑘‖𝐴_𝑘‖^{𝑞𝑔(𝐚,𝑝)})^1/𝑞<∞ and so on) with the usual modification for 𝑞=∞.

Using Theorems 4.5 and 3.8 in [8], we have the following.

Theorem 7. (1) Let 1<𝑝<∞. Then 𝛼∈𝑔(𝑝^∗), where 1/𝑝+1/𝑝^∗=1, if and only if [𝛼]∈(ℓ^∞𝑀(𝑝),ces^∞𝑀(𝑝)), where 𝑔(𝑝^∗)=𝑔(𝐚,𝑝^∗), with 𝐚=(1,1,…).
(2)  Let 0<𝑝<∞. Then [𝛼]∈(𝑑^∞𝑀(𝐚,𝑝),ℓ^∞𝑀(𝑝)) if and only if 𝛼∈𝑔(𝐚,𝑝).

Theorem 8. (1) For 1<𝑝<∞, [𝛼]∈(ℓ^𝑞𝑀(𝑝),ces^𝑞𝑀(𝑝)) if and only if 𝛼∈𝑔(𝑝^∗).
(2) For 0<𝑝<∞, [𝛼]∈(𝑑^𝑞𝑀(𝐚,𝑝),ℓ^𝑞𝑀(𝑝)) if and only if 𝛼∈𝑔(𝐚,𝑝).

Acknowledgments

This paper was partially supported by the CNCSIS Grant ID-PCE 1905/2008. The author thanks the referee for his/her valuable remarks which improved considerably the presentation of this paper.