A new class of linear operators on 2 and Schur multipliers for them

We introduce the space Bw(l2) of linear (unbounded) operators on l2 which map decreasing sequences from l2 into sequences from l2 and we find some classes of operators belonging either to Bw(l2) or to the space of all Schur multipliers on Bw(l2). For instance we show that the space B(l2) of all bounded operators on l2 is contained in the space of all Schur multipliers on Bw(l2).


Introduction
Let A = (a ij ) i,j≥1 be an infinite matrix.We define B w l 2 = A infinite matrix : Ax ∈ l 2 for every x ∈ l 2 with |x k | 0 , where The above space of matrices has appeared in the study of the matriceal analogue of some well-known Banach spaces as C (T), M (T), L 1 (T).
A similarity between the functions defined on T and the infinite matrices was remarked for the first time in 1978 by J. Arazy [1].Later on, in 1983, A. Shields has exploited further this similarity starting with a few constructs used in harmonic analysis together with their matriceal analogues [9].
Recently, in [3], the Fejer's theory developed for Fourier series was extended in the framework of matrices.
The analogy is as follows: we identify a function f with the Toeplitz matrix A = (a ij ) i,j≥1 , a ij = a i−j for all i, j ∈ N * where (a k ) k∈Z is the sequence of Fourier coefficients of the function f.
The Schur product of two matrices is defined by where A = (a ij ) i,j≥1 , B = (b ij ) i,j≥1 .We denote by M l 2 = M : M * A ∈ B l 2 for every A ∈ B l 2 the space of all Schur multipliers equiped with the following norm Let us define A k = (a ij ) i,j≥1 , k ∈ Z to be the matrix with the elements In this way we can say that B(l 2 ) represents the matriceal analogue of L ∞ (T) and M (l 2 ) is the analogue of M (T).Moreover in [3] was introduced the space of continuous matrices denoted by C(l 2 ).A is a continuous matrix if lim n→∞ σ n (A) = A, where the limit is taken in the norm of B(l 2 ) and σ n (A) is the Cesaro sum associated to S n (A) : This space is a Banach space with the following norm In the way described earlier this space represents the matriceal analogue of the space C(T).
Also, N. Popa has defined where This was the first attempt to define the matriceal analogue of the Wiener algebra A(T) and we may call it matriceal Wiener algebra.
= n, where e k = (0, ..., 0, 1, 0, ...) One solution would be to choose a larger space than B l 2 , and, for this reason we introduce B w l 2 .Clearly B w l 2 is a Banach space with the norm The following result due to E. Sawyer [8] (See also [7]) will be used very often in the sequel.
) and Ṽ (t) = t 0 ṽ(s)ds.Let us mention moreover that the relation f ≈ g means that there are two positive constants a and b such that af ≤ g ≤ bf.
Then we have ) n be weights on N * and let .
w(k) and similarly for V .
where 1 p + 1 p = 1.The paper is organised as follows: in Section 2, we show that the matriceal Wiener algebra A( 2) is a subset of B w ( 2 ) (Proposition 2) and we give some criteria for diagonal matrices to belong to B w ( 2 ).Moreover, we consider the Schur product of matrices and remark that B w ( 2 ) is not closed under this product.Using the Sawyer result we prove in Section 3 the main result of the paper, namely that linear and bounded operators on 2 are Schur multipliers on B w ( 2 ), a result which is not obvious, since B w ( 2 ) is not Schur algebra.Finally, we collected in Section 4 all results concerning the Toeplitz matrices.For instance there is no difference between Toeplitz matrices from B( 2) and those from B w ( 2 ).

Preliminary results
Proof.Let A ∈ A l 2 be an upper triangular matrix.Then for every Proof.The sufficiency follows immediately from the factorization see, e.g., [5].For the necessity take and sup If we translate this sequence (a k ) ∞ k=1 above or below of the main diagonal we obtain the following similar result.
1. Clearly B(l 2 ) ⊆ B w (l 2 ) and using Proposition 3 for the matrix we can easily show that the inclusion is proper.

While B(l 2
) is closed under Schur multiplication B w (l 2 ) is not.
For example it is easy to see that A * A / ∈ B w (l 2 ) where A is the matrix defined previously.
3. The space B w (l 2 ) cannot be compared with M (l 2 ) meaning that For (1) For (2) we take From Proposition 3, 2), provided in [5] we get that

Main result
Lemma 6. sup , where (a n ) n and (x n ) n are sequences of complex numbers.
Proof.We denote S = sup . From Theorem 1 we have , where a 0 = 0, therefore, for t ∈ (j, j + 1), we have On the other hand

Using now
Sawyer's formula for p = 1 we have sup .
Using Proposition 3 the matrix But sup < ∞ which completes the proof.
We remark here that if we translate this result to the case of functions we shall get a necessary condition for a function to belong to Proof.Let A be a Toeplitz matrix.Clearly, if A ∈ B l 2 it follows that A ∈ B w l 2 .It is well known that a Toeplitz matrix A = (a ij ), where a ij = a i−j for all i, j ∈ N * , maps l 2 into l 2 precisely when there exists a measurable function essentially bounded on [0, 2π] with Fourier coefficients f (n) = a n (n = 0, ±1, ±2, ...) and A B(l 2 ) = f ∞ see e.g.[11].
We follow the standard method [2] for solving the "problem of moments".Let p(t) = It is clear now that this map is well-defined and continuous and the existence of a measure satisfying all the requirements of the theorem follows easily from the Hahn-Banach and Riesz representation theorems.
Proof.This follows immediately from Riemann-Lebesgue lemma.

∈
B w l 2 if and only if sup

2 N<
∞, where f (k) is the Fourier coefficient of k -order.Theorem 9. B w l 2 ∩ T = B l 2 ∩ T , where T is the set of all Toeplitz matrices.