Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 142731, 11 pagesdoi:10.1155/2012/142731

Research Article

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

Nicolae Popa

Unit Research 1, Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Received 31 July 2011; Accepted 16 November 2011

Academic Editor: Lars Erik Persson

Copyright © 2012 Nicolae Popa. 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 characterize the Schur multipliers of scalar type acting on scattered classes of infinite matrices.

In [1], Schur introduced a new product between two matrices and 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

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 ,

In particular, he was interested in characterizing the -multipliers: the matrices for which maps into whenever 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 is a sequence of complex numbers.

Theorem 8.1 in [2] reads as follows.

Theorem B. A Toeplitz matrix is a Schur multiplier if and only if is a bounded Borel measure on .

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

The space endowed with norm becomes a Banach space.

Since we work with different quasi-Banach spaces of matrices we use the notation for the space of all Schur multipliers from into equipped with the quasi-norm

In this way becomes a quasi-Banach.

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

We recall that a matrix is called a Hankel matrix if it is defined by a sequence 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 maps boundedly into itself.

Here is the Hardy space of the Schatten class -valued analytic functions, endowed with the norm . For the definition of the Schatten classes , see, for example, [5].

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

Theorem AP. Let . A Toeplitz matrix given by the complex sequence belongs to if and only if there exists a measure with the Fourier coefficients . Moreover, in this case where , and is the Dirac measure concentrated at the point .

The above-mentioned papers [4, 5] show that a complete description of general Schur multipliers, at least, either for or 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 was introduced. Namely, we put and write .

Let , where , be the matrix given by

We call the matrix , the th corner matrix associated to .

Now, we associate to each matrix a periodical distribution on , denoted by , such that , and we identify the matrix with the sequence of associated distributions .

Then for the sequence and the matrix , we denote by the matrix given by .

In particular, if is a Toeplitz matrix and if is the constant sequence then coincides with the matrix .

Hence, if is the matrix it is clear that .

We define to be the space of all sequences such that for all , or equivalently .

On we consider the norm . Then 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   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 . For an infinite matrix the upper triangular projection is

A sequence belongs to if and only if

The space endowed with the norm 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 or , that is, to be Schur multipliers. The following result was proved.

Theorem BLP1. Let be a complex sequence. (1)If is a strictly increasing sequence of natural numbers with , and , then there is a constant such that (2)If then (3)If is a decreasing sequence, then if and only if .

As an immediate consequence we have the following.

Corollary 1. (1) One has . (2) One has .

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

Theorem BLP2. Let a complex sequence. Then, (1)if , then ;(2)if then .

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

Theorem 2. Let . Then .

Proof. By using the Schmidt decomposition of a matrix , it is enough to show that for a matrix of rank 1. Let with and .We have By the definition of and Cauchy-Schwartz inequality we get that is, and the proof is complete.

We characterize now the upper triangular scalar matrices which are Schur multipliers, from the Hardy space , respectively, from the Schatten class into .

Theorem 3. Let be the Hardy space of Toeplitz matrices generated by the classical Hardy space of functions. Then an upper triangular matrix belongs to if and only if . Moreover, one has equality of the norms. Let be the space of all upper triangular Hilbert-Schmidt matrices. Then if and only if .

Proof. (1) We use the following identity proved in [6]: where is an upper triangular matrix .Then, if , is the Toeplitz matrix associated to (i.e., is given by ), and , we have Hence , and this completes the proof. Let and Using formula (17) and Cauchy-Schwartz inequality we get Hence , and this proves the first part of the theorem.Conversely, let , that is, for all and take , that is, the matrix is reduced to main diagonal. It is clear that the sequence of entries of this diagonal belongs to . Consequently the sequence belongs to for every sequence . Hence , 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 . We denote these spaces by .

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 be the space of all sequences with a finite number of nonzero elements. A norm on is called symmetric if , for all , that is, if is invariant to permutations and to applications , where is a sequence of real numbers. Here is the decreasing rearrangement of the sequence which converges to 0.

We say that the sequence belongs to the space , if and only if exists.

We denote by the space of all compact operators on with the sequence of their singular numbers belonging to . For we put .

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

Theorem AH. Let be symmetric norms such that if then and If , then and .

Using this inequality we can state the following interesting result.

Theorem 4. Let (i.e., for each , there exist such that , and ). Then a scalar matrix if and only if .

Proof. Let first and . Then it is clear that where By Theorem AH it follows that Hence , and this completes the first part of the proof.For the reverse implication, take to be the main diagonal with the entries and .Then and we get that for all sequences . Since , it follows that , and this completes the proof of the theorem.

Let be a positive decreasing sequence of numbers. Of course the Lorentz space of sequences , , is a space of the previous type , see, for example, [9]. By the well-known fact that , for we get the following result.

Corollary 2. (1) Let . Then if and only if . (2) Let be a decreasing positive sequence, and let , , be such that . Then if and only if , where is the weighted Lorentz space of sequences.

We call the Bergman-Schatten space of order , , and we denote by the space of all upper triangular matrices such that . See, for example, [10] for further notations and details.

By Hölder’s inequality we get the following result.

Theorem 6. Let . Then if and only if , where .

Proof. Let and . We clearly have that . By Theorem AH we get that is, , and this completes the first part of the proof.Conversely, let . By taking , that is, for , we get or, equivalently, . Hence by Hölder’s inequality it follows that , 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 , , and were defined in [8].

We denote now by ,,, and the spaces of upper triangular infinite matrices , with all the sequences on the diagonals belonging to (resp., ,, ,) and such that (resp., and so on) with the usual modification for .

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

Theorem 7. (1) Let . Then , where , if and only if , where , with . (2)  Let . Then if and only if .

Theorem 8. (1) For , if and only if . (2) For , if and only if .

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.

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