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

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

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 products A * B a jk b 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 ≤ ∞, In particular, he was interested in characterizing the p, q -multipliers: the matrices M for which M * A maps p into 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 form 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 j e 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 2 the space of all 2,2 Schur multipliers from B 2 into B 2 , where B 2 is, as usual, the Banach space of linear and bounded operators on 2 with the usual operator norm.
The space M 2 endowed with norm 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 quasinorm 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: 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
Here H 1 S 1 is the Hardy space of the Schatten class S 1 -valued analytic functions, endowed with the norm 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 where M p {μ : T → C | μ j α j δ t j , t j ∈ T, t j distinct points}, μ M p 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 2 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 put

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 l k 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 α 1 , α 2 , . . .and the matrix B f l l∈N * , we denote by α B the matrix given by α l f 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 We define ms to be the space of all sequences α such that α B ∈ B 2 for all B ∈ B 2 , or equivalently α ∈ M 2 .
On ms we consider the norm α ms α M 2 .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 is The space pms endowed with the norm b {b} M 2 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, to be Schur multipliers.The following result was proved.
Theorem BLP 1.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 i 1 0, and As an immediate consequence we have the following.
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 BLP 2. Let b
b n n≥1 a complex sequence.Then, It is well known that M 2 coincides with S 1 , S 1 , the space of all Schur multipliers from S 1 into S 1 , see, for example, 4 .Using this fact we give a simpler proof of the first statement of Corollary 1.
Proof.By using the Schmidt decomposition of a matrix A, it is enough to show that A * {c} ∈ S 1 for a matrix A of rank 1.Let A α ⊗ β with α α n n≥1 ∈ 2 and β β n n≥1 ∈ 2 .
We have By the definition of S 1 and Cauchy-Schwartz inequality we get 2 and the proof is complete.We characterize now the upper triangular scalar matrices which are Schur multipliers, from the Hardy space H 2 , respectively, from the Schatten class S 2 into B 2 .

Theorem 3. 1 Let H 2 be the Hardy space of Toeplitz matrices generated by the classical Hardy space of functions. Then an upper triangular matrix A {α} belongs to H
, 17 where B is an upper triangular matrix B b kj .
Then, if f t ∞ k 0 c k e 2πikt ∈ H 2 , t ∈ 0, 1 , F is the Toeplitz matrix associated to f i.e., F is given by c k ∞ k≥0 , and α ∈ 2 , we have

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Hence {α} ∈ T 2 , B 2 , and this proves the first part of the theorem.Conversely, let {α} ∈ T 2 , B 2 , that is, C * {α} ∈ B 2 for all C ∈ T 2 and take C C 0 , that is, the matrix C is reduced to main diagonal.It is clear that the sequence of entries of this diagonal belongs to 2 .Consequently the sequence α k−1 c kk k≥0 belongs to ∞ for every sequence c kk ∞ k≥1 ∈ 2 .Hence α k ∞ k 0 ∈ ∞ , 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 iθ n a 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 a 1 , . . ., a n , 0, 0, . . .Φ a exists.We denote by S Φ the space of all compact operators A on 2 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.
Using this inequality we can state the following interesting result. where By Theorem AH it follows that Hence α ∈ LTS Φ 2 , LTS Φ 1 , 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 Then and we get that a ii α i , it follows that α ∈ s Φ 3 , 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 p,w , 0 < p ≤ ∞, is a space of the previous type s Φ , see, for example, 9 .By the well-known fact that p,w • q,w r,w , for 1/p 1/q 1/r; 0 < p, q, r < ∞ we get the following result.
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 ∈ q,w , where 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 p a 2 the space of all upper triangular matrices A such that < ∞.See, for example, 10 for further notations and details.
By H ölder's inequality we get the following result.
Proof.Let A ∈ L p a 2 and α ∈ q .We clearly have that A * α D α • A. By Theorem AH we get or, equivalently, α j a jj j ∈ 1 .Hence by H ölder's inequality it follows that α j j ∈ 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 q M a, p , g q M a, p , ces q M p , and q M p the spaces of upper triangular infinite matrices A ∞ k 0 A k , with all the sequences on the diagonals belonging to d a, p resp., g a, p , ces p , p , and such that A k A k q d a,p 1/q < ∞ resp., 1/q < ∞ and so on with the usual modification for q ∞.Using Theorems 4.5 and 3.8 in 8 , we have the following.

10 A
sequence b b n n≥1 belongs to pms if and only if 2 , B 2 if and only if α ∈ 2 .Moreover, one has equality of the norms. 2 Let T 2 be the space of all upper triangular Hilbert-Schmidt matrices.Then {α} ∈ T 2 , B 2 if and only if α ∈ ∞ .