Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volumeย 2012, Article IDย 142731, 11 pages
http://dx.doi.org/10.1155/2012/142731
Research Article

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

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 Erikย Persson

Copyright ยฉ 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 ๐ด=(๐‘Ž๐‘—๐‘˜) and ๐ต=(๐‘๐‘—๐‘˜) 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 products๎€ท๐‘Ž๐ดโˆ—๐ต=๐‘—๐‘˜๐‘๐‘—๐‘˜๎€ธ.(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 โ€–โ‹…โ€–๐‘,๐‘ž,1โ‰ค๐‘,๐‘žโ‰คโˆž,โ€–๐ดโ€–๐‘โ‹…๐‘ž=supโ€–๐‘ฅโ€–๐‘โ‰ค1โŽ›โŽœโŽœโŽ๎“๐‘—|||||๎“๐‘˜๐‘Ž๐‘—๐‘˜๐‘ฅ๐‘˜|||||๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž.(2)

In particular, he was interested in characterizing the (๐‘,๐‘ž)-multipliers: the matrices ๐‘€ for which ๐‘€โˆ—๐ด maps โ„“๐‘ into โ„“๐‘ž whenever ๐ด does.

In his paper it is proved a theorem about Schur multipliers which are Toeplitz matrices, that is about the matrices of the formโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž๐ด=0๐‘Ž1๐‘Ž2๐‘Ž3โ‹ฏ๐‘Žโˆ’1๐‘Ž0๐‘Ž1๐‘Ž2โ‹ฏ๐‘Žโˆ’2๐‘Žโˆ’1๐‘Ž0๐‘Ž1โ‹ฏ๐‘Žโˆ’3๐‘Žโˆ’2๐‘Žโˆ’1๐‘Ž0โ‹ฏโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑโ‹ฑ,(3) where (๐‘Ž๐‘—)โˆž๐‘—=โˆ’โˆž is a sequence of complex numbers.

Theorem 8.1 in [2] reads as follows.

Theorem B. A Toeplitz matrix ๐ด is a Schur multiplier if and only if โˆ‘๐œ‡=โˆž๐‘—=โˆ’โˆž๐‘Ž๐‘—๐‘’๐‘–๐‘—๐‘ก is a bounded Borel measure on [0,2๐œ‹).

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 ๐‘€(โ„“2) the space of all (2,2) Schur multipliers from ๐ต(โ„“2) into ๐ต(โ„“2), where ๐ต(โ„“2) is, as usual, the Banach space of linear and bounded operators on โ„“2 with the usual operator norm.

The space ๐‘€(โ„“2) endowed with norm โ€–๐ดโ€–๐‘€(โ„“2)=supโ€–๐ตโ€–2๐ต(โ„“)โ‰ค1โ€–๐ดโˆ—๐ตโ€–๐ต(โ„“2) becomes a Banach space.

Since we work with different quasi-Banach spaces of matrices ๐‘‹,๐‘Œ we use the notation (๐‘‹,๐‘Œ) for the space of all Schur multipliers from ๐‘‹ into ๐‘Œ equipped with the quasi-normโ€–๐ดโ€–(๐‘‹,๐‘Œ)=supโ€–๐ตโ€–๐‘Œโ‰ค1โ€–๐ดโˆ—๐ตโ€–๐‘‹.(4)

In this way (๐‘‹,๐‘Œ) becomes a quasi-Banach.

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

We recall that a matrix ๐ด is called a Hankel matrix if it is defined by a sequence (๐‘Ž๐‘—)โˆž๐‘—=1 of complex numbers in the following way:โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž๐ด=0๐‘Ž1๐‘Ž2๐‘Ž3โ‹ฏ๐‘Ž1๐‘Ž2๐‘Ž3๐‘Ž4โ‹ฏ๐‘Ž2๐‘Ž3๐‘Ž4๐‘Ž5โ‹ฏ๐‘Ž3๐‘Ž4๐‘Ž5๐‘Ž6โ‹ฏโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฎโ‹ฎโ‹ฎ.(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 โˆ‘โˆž๐‘›=0๐‘ฅ๐‘›๐‘’intโ†’โˆ‘โˆž๐‘›=0๐‘Ž๐‘›๐‘ฅ๐‘›๐‘’int maps boundedly ๐ป1(๐‘†1) into itself.

Here ๐ป1(๐‘†1) is the Hardy space of the Schatten class ๐‘†1-valued analytic functions, endowed with the norm โ€–๐‘“โ€–๐ป1(๐‘†1)โˆซ=(1/2๐œ‹)02๐œ‹โ€–โˆ‘โˆž๐‘›=0๐ด๐‘›๐‘’intโ€–๐‘†1๐‘‘๐‘ก<โˆž. For the definition of the Schatten classes ๐‘†๐‘, see, for example, [5].

In [5], Aleksandrov and Peller characterized the Toeplitz matrices which are Schur multipliers for ๐‘†๐‘, 0<๐‘<1. They proved the following theorem.

Theorem AP. Let 0<๐‘<1. A Toeplitz matrix ๐‘‡ given by the complex sequence (๐‘ก๐‘—)โˆž๐‘—=โˆ’โˆž belongs to (๐‘†๐‘,๐‘†๐‘) if and only if there exists a measure ๐œ‡โˆˆ๐‘€๐‘ with the Fourier coefficients ๎๐œ‡(๐‘—)=๐‘ก๐‘—. Moreover, in this case โ€–๐‘‡โ€–(๐‘†๐‘,๐‘†๐‘)=โ€–๐œ‡โ€–๐‘€๐‘,(6) where ๐‘€๐‘โˆ‘={๐œ‡โˆถ๐•‹โ†’โ„‚โˆฃ๐œ‡=๐‘—๐›ผ๐‘—๐›ฟ๐‘ก๐‘—,๐‘ก๐‘—โˆˆ๐•‹,๐‘ก๐‘—๐‘‘๐‘–๐‘ ๐‘ก๐‘–๐‘›๐‘๐‘ก๐‘๐‘œ๐‘–๐‘›๐‘ก๐‘ },โ€–๐œ‡โ€–๐‘€๐‘โˆ‘=(๐‘—|๐›ผ๐‘—|๐‘)1/๐‘<โˆž, and ๐›ฟ๐‘ก is the Dirac measure concentrated at the point ๐‘กโˆˆ๐•‹.

The above-mentioned papers [4, 5] show that a complete description of general Schur multipliers, at least, either for ๐ต(โ„“2) or ๐‘†๐‘,0<๐‘โ‰ค1, 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 ๐ต was introduced. Namely, we put๐‘๐‘™๐‘˜=๎‚ป๐‘๐‘™,๐‘™+๐‘˜๐‘,๐‘˜โ‰ฅ0,๐‘™=1,2,โ€ฆ,๐‘™โˆ’๐‘˜,๐‘™,๐‘˜<0,๐‘™=1,2,โ€ฆ,(7) and write ๐ต=(๐‘๐‘™๐‘˜)๐‘™โ‰ฅ1,๐‘˜โˆˆโ„ค.

Let ๐ต(๐‘™)=(๐‘๐‘š๐‘˜)๐‘˜โˆˆโ„ค,๐‘šโ‰ฅ1, where ๐‘™=1,2,3,โ€ฆ, be the matrix given by๐‘๐‘š๐‘˜=๎‚ป๐‘๐‘™๐‘˜,if๐‘š=๐‘™,0,if๐‘šโ‰ ๐‘™.(8)

We call the matrix ๐ต(๐‘™), the ๐‘™th corner matrix associated to ๐ต.

Now, we associate to each matrix ๐ต๐‘™ a periodical distribution on ๐•‹, denoted by ๐‘“๐‘™, such that ๐‘๐‘™๐‘˜=๎๐‘“๐‘™(๐‘˜), and we identify the matrix ๐ต=(๐ต(๐‘™))๐‘™โˆˆโ„•โˆ— with the sequence of associated distributions (๐‘“๐‘™)๐‘™โˆˆโ„•โˆ—.

Then for the sequence ๐›ผ=(๐›ผ1,๐›ผ2,โ€ฆ) and the matrix ๐ต=(๐‘“๐‘™)๐‘™โˆˆโ„•โˆ—, we denote by ๐›ผโŠ™๐ต the matrix given by (๐›ผ๐‘™๐‘“๐‘™)๐‘™โˆˆโ„•โˆ—.

In particular, if ๐ต is a Toeplitz matrix (๐ตโˆˆ๐’ฏ) and if ๐›ผ is the constant sequence then ๐›ผโŠ™๐ต coincides with the matrix ๐›ผ๐ต.

Hence, if [๐›ผ] is the matrix[๐›ผ]=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐›ผ1๐›ผ1๐›ผ1โ‹ฏ๐›ผ1๐›ผ2๐›ผ2โ‹ฑ๐›ผ1๐›ผ2๐›ผ3โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑ(9) it is clear that ๐›ผโŠ™๐ต=[๐›ผ]โˆ—๐ต.

We define ๐‘š๐‘  to be the space of all sequences ๐›ผ such that ๐›ผโŠ™๐ตโˆˆ๐ต(โ„“2) for all ๐ตโˆˆ๐ต(โ„“2), or equivalently [๐›ผ]โˆˆ๐‘€(โ„“2).

On ๐‘š๐‘  we consider the norm โ€–๐›ผโ€–๐‘š๐‘ =โ€–[๐›ผ]โ€–๐‘€(โ„“2). Then ๐‘š๐‘  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 โŠ™ of a sequence and an infinite matrix.

We call the matrices [๐›ผ]โ€‰โ€‰scalar matrices. In this context, in [6] a theorem of Haarโ€™s type for infinite matrices was proved. The product โŠ™ appeared also in [7] in other contexts.

An important role in applications is played by the upper triangular projection applied to the matrix [๐›ผ]. For an infinite matrix ๐ด=(๐‘Ž๐‘–๐‘—)๐‘–โ‰ฅ1,๐‘—โ‰ฅ1, the upper triangular projection is๐‘ƒ๐‘‡๎‚ป๐‘Ž(๐ด)=๐‘–,๐‘—,if๐‘–โ‰ค๐‘—,0,otherwise.(10)

A sequence ๐‘=(๐‘๐‘›)๐‘›โ‰ฅ1 belongs to ๐‘๐‘š๐‘  if and only if๐ต={๐‘}=๐‘ƒ๐‘‡([๐‘]๎€ทโ„“)โˆˆ๐‘€2๎€ธ.(11)

The space ๐‘๐‘š๐‘  endowed with the norm โ€–๐‘โ€–=โ€–{๐‘}โ€–๐‘€(โ„“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 [๐›ผ] or {๐›ผ}, that is,โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐›ผ{๐›ผ}=1๐›ผ1๐›ผ1โ‹ฏ0๐›ผ2๐›ผ2โ‹ฑ00๐›ผ3โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑ,(12) to be Schur multipliers. The following result was proved.

Theorem BLP1. Let ๐‘=(๐‘๐‘›)๐‘›โ‰ฅ1 be a complex sequence. (1)If (๐‘–๐‘›)๐‘›โ‰ฅ1 is a strictly increasing sequence of natural numbers with ๐‘–1=0, and ๐‘ง๐‘–๐‘›=max๐‘–๐‘›<๐‘˜โ‰ค๐‘–๐‘›+1|๐‘๐‘˜|, then there is a constant ๐‘…>0 such thatโ€–โ€–{๐‘}๐‘€(โ„“2)โ‰ค๐‘…inf(๐‘–๐‘›)๐‘›โ‰ฅ1๎‚†โ€–โ€–๎€ท๐‘ง๐‘–๐‘›๎€ธ๐‘›โ‰ฅ1โ€–โ€–2+โ€–โ€–๎€ท๐‘ง๐‘–๐‘›๎€ท๐‘–log๐‘›+1โˆ’๐‘–๐‘›๎€ธ๎€ธ๐‘›โ€–โ€–โˆž๎‚‡.(13)(2)If ๐‘โˆˆ๐‘๐‘š๐‘  thensup๐‘›โ‰ฅ1;๐‘โ‰ฅ1(log๐‘›)2๐‘›๐‘›+๐‘๎“๐‘˜=๐‘||๐‘๐‘˜||2<โˆž.(14)(3)If (|๐‘๐‘˜|)๐‘˜โ‰ฅ1 is a decreasing sequence, then ๐‘โˆˆ๐‘๐‘š๐‘  if and only if |๐‘๐‘˜|=๐’ช(1/log๐‘˜).

As an immediate consequence we have the following.

Corollary 1. (1) One has โ„“2โŠ‚๐‘š๐‘ โŠ‚โ„“โˆž.
(2) One has {(๐‘๐‘›)๐‘›โ‰ฅ1โˆฃ|๐‘๐‘›|=๐’ช(1/log๐‘›)}โŠ‚๐‘š๐‘ .

A set of sufficient conditions in order for a matrix of the type [๐›ผ] to be a Schur multiplier is given in [6], namely, the following theorem was proved.

Theorem BLP2. Let ๐‘=(๐‘๐‘›)๐‘›โ‰ฅ1 a complex sequence. Then, (1)if sup๐‘›โ‰ฅ1โˆ‘๐‘›๐‘—=1|๐‘๐‘—โˆ’๐‘๐‘›|2<โˆž, then ๐‘โˆˆ๐‘š๐‘ ;(2)if โ€–๐‘โ€–๐ต๐‘‰โ„•=|๐‘1โˆ‘|+โˆž๐‘›=1|๐‘๐‘›+1โˆ’๐‘๐‘›|<โˆž then ๐‘โˆˆ๐‘š๐‘ .

It is well known that ๐‘€(โ„“2) coincides with (๐‘†1,๐‘†1), the space of all Schur multipliers from ๐‘†1 into ๐‘†1, see, for example, [4]. Using this fact we give a simpler proof of the first statement of Corollary 1.

Theorem 2. Let ๐‘=(๐‘๐‘›)๐‘›โ‰ฅ1โˆˆโ„“2. Then ๐‘โˆˆ๐‘๐‘š๐‘ .

Proof. By using the Schmidt decomposition of a matrix ๐ด, it is enough to show that ๐ดโˆ—{๐‘}โˆˆ๐‘†1 for a matrix ๐ด of rank 1. Let ๐ด=๐›ผโŠ—๐›ฝ with ๐›ผ=(๐›ผ๐‘›)๐‘›โ‰ฅ1โˆˆโ„“2 and ๐›ฝ=(๐›ฝ๐‘›)๐‘›โ‰ฅ1โˆˆโ„“2.
We have โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐›ผ๐ดโˆ—{๐‘}=1๐›ฝ1๐›ผ1๐›ฝ2๐›ผ1๐›ฝ3โ‹ฏ๐›ผ2๐›ฝ1๐›ผ2๐›ฝ2๐›ผ2๐›ฝ3โ‹ฏ๐›ผ3๐›ฝ1๐›ผ3๐›ฝ2๐›ผ3๐›ฝ3โ‹ฏโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โˆ—โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘โ‹ฎโ‹ฎโ‹ฎ1๐‘1๐‘1๐‘1โ‹ฏ0๐‘2๐‘2๐‘2โ‹ฑ00๐‘3๐‘3โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘โ‹ฎโ‹ฑโ‹ฑโ‹ฑโ‹ฑ1๐›ผ1๐›ฝ1๐‘1๐›ผ1๐›ฝ2๐‘1๐›ผ1๐›ฝ3โ‹ฏ0๐‘2๐›ผ2๐›ฝ2๐‘2๐›ผ2๐›ฝ3โ‹ฏ00๐‘3๐›ผ3๐›ฝ3โ‹ฏโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .โ‹ฎโ‹ฎโ‹ฎ(15)
By the definition of ๐‘†1 and Cauchy-Schwartz inequality we get โ€–๐ดโˆ—{๐‘}โ€–๐‘†1โ‰คโˆž๎“๐‘—=1โ€–โ€–๎€ท๐›ผ๐‘—๐‘๐‘—๐›ฝ๐‘—+๐‘˜๎€ธโˆž๐‘˜=0โ€–โ€–โ„“2=โˆž๎“๐‘—=1๎ƒฉโˆž๎“๐‘˜=0||๐›ผ๐‘—๐‘๐‘—||2||๐›ฝ๐‘—+๐‘˜||2๎ƒช1/2โ‰ค๎ƒฉโˆž๎“๐‘—=1||๐›ผ๐‘—๐‘๐‘—||๎ƒชโ€–โ€–๐›ฝ๐‘—โ€–โ€–โ„“2โ‰คโ€–๐›ผโ€–โ„“2โ€–๐›ฝโ€–โ„“2โ€–๐‘โ€–โ„“2=โ€–๐‘โ€–โ„“2โ€–๐ดโ€–๐‘†1,(16) that is, โ€–๐ดโ€–๐‘€(โ„“2)โ‰คโ€–๐‘โ€–โ„“2 and the proof is complete.

We characterize now the upper triangular scalar matrices which are Schur multipliers, from the Hardy space ๐ป2, respectively, from the Schatten class ๐‘†2 into ๐ต(โ„“2).

Theorem 3. (1) Let ๐ป2 be the Hardy space of Toeplitz matrices generated by the classical Hardy space of functions. Then an upper triangular matrix ๐ด={๐›ผ} belongs to (๐ป2,๐ต(โ„“2)) if and only if ๐›ผโˆˆโ„“2. Moreover, one has equality of the norms.
(2) Let ๐‘‡2 be the space of all upper triangular Hilbert-Schmidt matrices. Then {๐›ผ}โˆˆ(๐‘‡2,๐ต(โ„“2)) if and only if ๐›ผโˆˆโ„“โˆž.

Proof. (1) We use the following identity proved in [6]: โ€–๐ตโ€–๐ต(โ„“2)=supโ€–โ„Žโ€–2โ‰ค1;โ„Žโˆˆ๐ป20[]0,1โŽ›โŽœโŽœโŽโˆž๎“๐‘˜=1|||||๎€œ10โˆž๎“๐‘—=๐‘˜๐‘๐‘˜๐‘—๐‘’2๐œ‹๐‘–๐‘—๐‘กโ„Ž|||||(โˆ’๐‘ก)๐‘‘๐‘ก2โŽžโŽŸโŽŸโŽ 1/2,(17) where ๐ต is an upper triangular matrix ๐ต=(๐‘๐‘˜๐‘—).
Then, if โˆ‘๐‘“(๐‘ก)=โˆž๐‘˜=0๐‘๐‘˜๐‘’2๐œ‹๐‘–๐‘˜๐‘กโˆˆ๐ป2,๐‘กโˆˆ[0,1], ๐น is the Toeplitz matrix associated to ๐‘“ (i.e., ๐น is given by (๐‘๐‘˜)โˆž๐‘˜โ‰ฅ0), and ๐›ผโˆˆโ„“2, we have โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐›ผ๐นโˆ—{๐›ผ}=0๐‘0๐›ผ0๐‘1๐›ผ0๐‘2๐›ผ0๐‘3โ‹ฏ0๐›ผ1๐‘0๐›ผ1๐‘1๐›ผ1๐‘2โ‹ฑ00๐›ผ2๐‘0๐›ผ2๐‘1โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑโ‹ฑโ€–๐นโˆ—{๐›ผ}โ€–๐ต(โ„“2)=supโ€–โ„Žโ€–2โ‰ค1โŽ›โŽœโŽœโŽโˆž๎“๐‘˜=1|||||๎€œ10๐›ผ๐‘˜โˆ’1๎ƒฉโˆž๎“๐‘—=0๐‘๐‘—๐‘’2๐œ‹๐‘–๐‘—๐‘ก๎ƒช|||||โ„Ž(โˆ’๐‘ก)๐‘‘๐‘ก2โŽžโŽŸโŽŸโŽ 1/2=supโ€–โ„Žโ€–2โ‰ค1๎ƒฉโˆž๎“๐‘˜=1||๐›ผ๐‘˜โˆ’1||2๎ƒช1/2|||||๎€œ10๎ƒฉโˆž๎“๐‘—=0๐‘๐‘—๐‘’2๐œ‹๐‘–๐‘—๐‘ก๎ƒช|||||โ„Ž(โˆ’๐‘ก)๐‘‘๐‘ก=โ€–๐›ผโ€–โ„“2โ€–๐‘“โ€–๐ป2.(18)
Hence โ€–{๐›ผ}โ€–(๐ป2,๐ต(โ„“2))=โ€–๐›ผโ€–โ„“2, and this completes the proof.
(2) Let ๐›ผโˆˆโ„“โˆž and โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘๐ถ=11๐‘12๐‘13โ‹ฏ0๐‘22๐‘23โ‹ฏ00๐‘33โ‹ฏโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฎโ‹ฎโˆˆ๐‘‡2.(19) Using formula (17) and Cauchy-Schwartz inequality we get โ€–โ€–{๐›ผ}โˆ—๐ถ๐ต(โ„“2)โ‰คโŽ›โŽœโŽœโŽโˆž๎“๐‘˜=1||๐›ผ๐‘˜โˆ’1||2supโ€–โ„Žโ€–2โ‰ค1|||||๎€œ10๎ƒฉ๎“๐‘—โ‰ฅ๐‘˜๐‘๐‘˜๐‘—๐‘’2๐œ‹๐‘–๐‘—๐‘ก๎ƒช|||||โ„Ž(โˆ’๐‘ก)2โŽžโŽŸโŽŸโŽ 1/2โ‰ค๎ƒฉโˆž๎“๐‘˜=1||๐›ผ๐‘˜โˆ’1||2๎ƒฉ๎“๐‘—โ‰ฅ๐‘˜||๐‘๐‘˜๐‘—||2๎ƒช๎ƒช1/2โ‰ค๎‚ตsup๐‘˜โ‰ฅ0||๐›ผ๐‘˜||๎‚ถ๎ƒฉโˆž๎“๐‘˜=1๎“๐‘—โ‰ฅ๐‘˜||๐‘๐‘˜๐‘—||2๎ƒช1/2=โ€–๐›ผโ€–โ„“โˆžโ€–๐ถโ€–๐‘‡2.(20)
Hence {๐›ผ}โˆˆ(๐‘‡2,๐ต(โ„“2)), and this proves the first part of the theorem.
Conversely, let {๐›ผ}โˆˆ(๐‘‡2,๐ต(โ„“2)), that is, ๐ถโˆ—{๐›ผ}โˆˆ๐ต(โ„“2) for all ๐ถโˆˆ๐‘‡2 and take ๐ถ=๐ถ0, that is, the matrix ๐ถ is reduced to main diagonal. It is clear that the sequence of entries of this diagonal belongs to โ„“2. Consequently the sequence (๐›ผ๐‘˜โˆ’1๐‘๐‘˜๐‘˜)๐‘˜โ‰ฅ0 belongs to โ„“โˆž for every sequence (๐‘๐‘˜๐‘˜)โˆž๐‘˜โ‰ฅ1โˆˆโ„“2. Hence (๐›ผ๐‘˜)โˆž๐‘˜=0โˆˆโ„“โˆž, 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 ๐‘†๐‘,0<๐‘<โˆž. We denote these spaces by โ„’๐’ฏ๐‘†๐‘.

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 ๐Ÿ be the space of all sequences with a finite number of nonzero elements. A norm ฮฆ on ๐Ÿ is called symmetric if ฮฆ(๐‘Ž)=ฮฆ(๐‘Žโˆ—), for all ๐‘Žโˆˆ๐Ÿ, that is, if ฮฆ is invariant to permutations and to applications ๐‘Ž๐‘›โ†’๐‘’๐‘–๐œƒ๐‘›๐‘Ž๐‘›, where ๐œƒ๐‘› is a sequence of real numbers. Here ๐‘Žโˆ—=(๐‘Žโˆ—๐‘›)โˆž๐‘›=1 is the decreasing rearrangement of the sequence (๐‘Ž๐‘›) which converges to 0.

We say that the sequence (๐‘Ž๐‘›)๐‘› belongs to the space ๐‘ ฮฆ, if and only if lim๐‘›โ†’โˆžฮฆ(๐‘Ž1,โ€ฆ,๐‘Ž๐‘›,0,0,โ€ฆ)=ฮฆ(๐‘Ž) exists.

We denote by ๐‘†ฮฆ the space of all compact operators ๐ด on โ„“2 with the sequence of their singular numbers (๐œ‡๐‘›(๐ด)) belonging to ๐‘ ฮฆ. For ๐ดโˆˆ๐‘†ฮฆ we put ฮฆ(๐ด)=ฮฆ((๐œ‡๐‘›(๐ด))๐‘›).

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

Theorem AH. Let ฮฆ1,ฮฆ2,ฮฆ3 be symmetric norms such that if ๐‘Žโˆˆ๐‘ ฮฆ2,๐‘โˆˆ๐‘ ฮฆ3 then ๐‘Ž๐‘โˆˆ๐‘ ฮฆ1 and ฮฆ1(๐‘Ž๐‘)โ‰คฮฆ2(๐‘Ž)ฮฆ3(๐‘).(21) If ๐ดโˆˆ๐‘†ฮฆ2,๐ตโˆˆ๐‘†ฮฆ3, then ๐ด๐ตโˆˆ๐‘†ฮฆ1 and ฮฆ1(๐ด๐ต)โ‰คฮฆ2(๐ด)ฮฆ3(๐ต).

Using this inequality we can state the following interesting result.

Theorem 4. Let ๐‘ ฮฆ1=๐‘ ฮฆ2๐‘ ฮฆ3 (i.e., for each ๐›ผโˆˆ๐‘ ฮฆ1, there exist ๐›ฝโˆˆ๐‘ ฮฆ2,๐›พโˆˆ๐‘ ฮฆ3 such that ๐›ผ=๐›ฝ๐›พ, and ฮฆ1(๐›ผ)โ‰ˆinf๐›ผ=๐›ฝ๐›พฮฆ2(๐›ฝ)ฮฆ3(๐›พ)). Then a scalar matrix [๐›ผ]โˆˆ(โ„’๐’ฏ๐‘†ฮฆ2,โ„’๐’ฏ๐‘†ฮฆ1) if and only if ๐›ผโˆˆ๐‘ ฮฆ3.

Proof. Let first ๐ดโˆˆโ„’๐’ฏ๐‘†ฮฆ1 and ๐›ผโˆˆ๐‘ ฮฆ3. Then it is clear that [๐›ผ]๐ดโˆ—=๐ดโ‹…๐ท๐›ผ,(22) where ๐ท๐›ผ=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐›ผ100โ‹ฏ0๐›ผ20โ‹ฑ00๐›ผ3โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑ.(23)
By Theorem AH it follows that [๐›ผ]โ€–โ€–๐ดโˆ—๐‘†ฮฆ1โ‰คโ€–โ€–๐ดโ‹…๐ท๐›ผโ€–โ€–๐‘†ฮฆ1โ‰คโ€–๐ดโ€–๐‘†ฮฆ2โ€–โ€–๐ท๐›ผโ€–โ€–๐‘†ฮฆ3โ‰คโ€–๐ดโ€–๐‘†ฮฆ2โ€–๐›ผโ€–๐‘ ฮฆ3.(24) Hence [๐›ผ]โˆˆ(โ„’๐’ฏ๐‘†ฮฆ2,โ„’๐’ฏ๐‘†ฮฆ1), and this completes the first part of the proof.
For the reverse implication, take ๐ด to be the main diagonal with the entries (๐‘Ž๐‘—๐‘—)โˆž๐‘—=1โˆˆ๐‘ ฮฆ2 and [๐›ผ]โˆˆ(โ„’๐’ฏ๐‘†ฮฆ2,โ„’๐’ฏ๐‘†ฮฆ1).
Then [๐›ผ]๐ดโˆ—=๐ดโ‹…๐ท๐›ผ=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž11๐›ผ100โ‹ฏ0๐‘Ž22๐›ผ20โ‹ฑ00๐‘Ž33๐›ผ3โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโ‹ฑโˆˆ๐‘†ฮฆ1(25) and we get that (๐‘Ž๐‘–๐‘–๐›ผ๐‘–)โˆž๐‘–=1โˆˆ๐‘ ฮฆ1 for all sequences (๐‘Ž๐‘—๐‘—)โˆž๐‘—=1โˆˆ๐‘ ฮฆ2. Since ๐‘ ฮฆ1=๐‘ ฮฆ2๐‘ ฮฆ3, it follows that ๐›ผโˆˆ๐‘ ฮฆ3, and this completes the proof of the theorem.

Let ๐‘ค=(๐‘ค๐‘›) be a positive decreasing sequence of numbers. Of course the Lorentz space of sequences โ„“๐‘,๐‘ค, 0<๐‘โ‰คโˆž, is a space of the previous type ๐‘ ฮฆ, see, for example, [9]. By the well-known fact that โ„“๐‘,๐‘คโ‹…โ„“๐‘ž,๐‘ค=โ„“๐‘Ÿ,๐‘ค, for 1/๐‘+1/๐‘ž=1/๐‘Ÿ;0<๐‘,๐‘ž,๐‘Ÿ<โˆž we get the following result.

Corollary 2. (1) Let 1/๐‘+1/๐‘ž=1/๐‘Ÿ,0<๐‘,๐‘ž,๐‘Ÿ<โˆž. Then [๐›ผ]โˆˆ(๐‘†๐‘,๐‘†๐‘Ÿ) if and only if ๐›ผโˆˆโ„“๐‘ž.
(2) Let ๐‘ค๐‘› be a decreasing positive sequence, and let 0<๐‘, ๐‘ž<โˆž, be such that 1/๐‘+1/๐‘ž=1. Then [๐›ผ]โˆˆ(๐‘†๐‘,๐‘ค,๐‘†1,๐‘ค) if and only if ๐›ผโˆˆโ„“๐‘ž,๐‘ค, where โ„“๐‘,๐‘ค is the weighted Lorentz space of sequences.

We call the Bergman-Schatten space of order ๐‘, 0<๐‘<โˆž, and we denote by ๐ฟ๐‘๐‘Ž(โ„“2) the space of all upper triangular matrices ๐ด such that โ€–๐ดโ€–๐ฟ๐‘๐‘Ž(โ„“2)โˆซ=(10โ€–โˆ‘โˆž๐‘˜=0๐ด๐‘˜๐‘Ÿ๐‘˜โ€–๐‘๐‘†๐‘2๐‘Ÿ๐‘‘๐‘Ÿ)1/๐‘<โˆž. See, for example, [10] for further notations and details.

By Hรถlderโ€™s inequality we get the following result.

Theorem 6. Let 1โ‰ค๐‘<โˆž. Then [๐›ผ]โˆˆ(๐ฟ๐‘๐‘Ž(โ„“2),๐ฟ1๐‘Ž(โ„“2)) if and only if ๐›ผโˆˆโ„“๐‘ž, where 1/๐‘+1/๐‘ž=1.

Proof. Let ๐ดโˆˆ๐ฟ๐‘๐‘Ž(โ„“2) and ๐›ผโˆˆโ„“๐‘ž. We clearly have that ๐ดโˆ—[๐›ผ]=๐ท๐›ผโ‹…๐ด. By Theorem AH we get [๐›ผ]โ€–โ€–๐ดโˆ—๐ฟ1๐‘Ž(โ„“2)=๎€œ10โ€–โ€–๐ท๐›ผโ€–โ€–โ‹…๐ด(๐‘Ÿ)๐‘†1โ‰ค๎‚ต๎€œ2๐‘Ÿ๐‘‘๐‘Ÿ10โ€–๐ด(๐‘Ÿ)โ€–๐‘๐‘†๐‘๎‚ถ2๐‘Ÿ๐‘‘๐‘Ÿ1/๐‘โ€–๐›ผโ€–โ„“๐‘ž=โ€–๐ดโ€–๐ฟ๐‘๐‘Ž(โ„“2)โ€–๐›ผโ€–โ„“๐‘ž,(26) that is, [๐›ผ]โˆˆ(๐ฟ๐‘๐‘Ž(โ„“2),๐ฟ1๐‘Ž(โ„“2)), and this completes the first part of the proof.
Conversely, let [๐›ผ]โˆˆ(๐ฟ๐‘๐‘Ž(โ„“2),๐ฟ1๐‘Ž(โ„“2)). By taking ๐ด=๐ด0=(๐‘Ž๐‘—๐‘—)โˆž๐‘—=1โˆˆ๐ฟ๐‘๐‘Ž(โ„“2), that is, for (๐‘Ž๐‘—๐‘—)๐‘—โˆˆโ„“๐‘, we get [๐›ผ]=โŽ›โŽœโŽœโŽœโŽœโŽ๐›ผ๐ดโˆ—1๐‘Ž110โ‹ฏ0๐›ผ2๐‘Ž22โ‹ฑโŽžโŽŸโŽŸโŽŸโŽŸโŽ โ‹ฎโ‹ฑโ‹ฑโˆˆ๐ฟ1๐‘Ž๎€ทโ„“2๎€ธ,(27) or, equivalently, (๐›ผ๐‘—๐‘Ž๐‘—๐‘—)๐‘—โˆˆโ„“1. Hence by Hรถlderโ€™s inequality it follows that (๐›ผ๐‘—)๐‘—โˆˆโ„“๐‘ž, 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 ๐‘‘(๐š,๐‘), ๐‘”(๐š,๐‘), and ces(๐‘) were defined in [8].

We denote now by ๐‘‘๐‘ž๐‘€(๐š,๐‘),๐‘”๐‘ž๐‘€(๐š,๐‘),ces๐‘ž๐‘€(๐‘), and โ„“๐‘ž๐‘€(๐‘) the spaces of upper triangular infinite matrices โˆ‘๐ด=โˆž๐‘˜=0๐ด๐‘˜, with all the sequences on the diagonals belonging to ๐‘‘(๐š,๐‘) (resp., ๐‘”(๐š,๐‘),ces(๐‘), โ„“๐‘,) and such that โˆ‘โ€–๐ดโ€–=(๐‘˜โ€–๐ด๐‘˜โ€–๐‘ž๐‘‘(๐š,๐‘))1/๐‘ž<โˆž (resp., โˆ‘โ€–๐ดโ€–=(๐‘˜โ€–๐ด๐‘˜โ€–๐‘ž๐‘”(๐š,๐‘))1/๐‘ž<โˆž and so on) with the usual modification for ๐‘ž=โˆž.

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

Theorem 7. (1) Let 1<๐‘<โˆž. Then ๐›ผโˆˆ๐‘”(๐‘โˆ—), where 1/๐‘+1/๐‘โˆ—=1, if and only if [๐›ผ]โˆˆ(โ„“โˆž๐‘€(๐‘),cesโˆž๐‘€(๐‘)), where ๐‘”(๐‘โˆ—)=๐‘”(๐š,๐‘โˆ—), with ๐š=(1,1,โ€ฆ).
(2)โ€‰โ€‰Let 0<๐‘<โˆž. Then [๐›ผ]โˆˆ(๐‘‘โˆž๐‘€(๐š,๐‘),โ„“โˆž๐‘€(๐‘)) if and only if ๐›ผโˆˆ๐‘”(๐š,๐‘).

Theorem 8. (1) For 1<๐‘<โˆž, [๐›ผ]โˆˆ(โ„“๐‘ž๐‘€(๐‘),ces๐‘ž๐‘€(๐‘)) if and only if ๐›ผโˆˆ๐‘”(๐‘โˆ—).
(2) For 0<๐‘<โˆž, [๐›ผ]โˆˆ(๐‘‘๐‘ž๐‘€(๐š,๐‘),โ„“๐‘ž๐‘€(๐‘)) if and only if ๐›ผโˆˆ๐‘”(๐š,๐‘).

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.

References

  1. J. Schur, โ€œBemerkungen zur Theorie der beschrรคnkten Bilinearformen mit unendlich vielen Verรคnderlichen,โ€ Journal fur die Reine und Angewandte Mathematik, vol. 140, pp. 1โ€“28, 1911. View at Google Scholar
  2. G. Bennett, โ€œSchur multipliers,โ€ Duke Mathematical Journal, vol. 44, no. 3, pp. 603โ€“639, 1977. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  3. S. Barza, L.-E. Persson, and N. Popa, โ€œA matriceal analogue of Fejer's theory,โ€ Mathematische Nachrichten, vol. 260, pp. 14โ€“20, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  4. G. Pisier, Similarity Problems and Completely Bounded Maps, vol. 1618 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1995.
  5. A. B. Aleksandrov and V. V. Peller, โ€œHankel and Toeplitz-Schur multipliers,โ€ Mathematische Annalen, vol. 324, no. 2, pp. 277โ€“327, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  6. S. Barza, V. Lie, and N. Popa, โ€œApproximation of infinite matrices by matricial Haar polynomials,โ€ Arkiv fรถr Matematik, vol. 43, no. 2, pp. 251โ€“269, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  7. A. Marcoci, L. E. Persson, and N. Popa, โ€œSchur multiplier characterization of a class of infinite matrices,โ€ Czechoslovak Mathematical Journal, vol. 60, no. 1, pp. 183โ€“193, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  8. G. Bennett, โ€œFactorizing the classical inequalities,โ€ Memoirs of the American Mathematical Society, vol. 120, no. 576, pp. 1โ€“130, 1996. View at Google Scholar ยท View at Zentralblatt MATH
  9. B. Simon, Trace Ideals and Their Applications, vol. 35 of Lecture Notes in Mathematics, Cambridge University Press, Cambridge, UK, 1979.
  10. L. G. Marcoci, L. E. Persson, I. Popa, and N. Popa, โ€œA new characterization of Bergman-Schatten spaces and a duality result,โ€ Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 67โ€“80, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH