Matriceal Lebesgue spaces and Hölder inequality

We introduce a class of spaces of infinite matrices similar to the class of Lebesgue spaces L (T) , 1 ≤ p ≤ ∞, and we prove matriceal versions of Hölder inequality.


Introduction
Let A = (a i,j ) i,j≥1 be an infinite complex matrix.We denote by A k , k ∈ Z, the diagonal matrix whose entries a i,j , satisfy the equation: It was remarked first in [1] that there is a similarity between the Fourier series k∈Z a k e ikt of a function on the torus T = [0, 2π) and the decomposition k∈Z A k of an infinite matrix in a sum of diagonal matrices.
Moreover Shields [8] observed the analogy between the convolution of two functions and the Schur product A * B = (a ij • b ij ) i,j of the matrices A = (a ij ) i,j and B = (b ij ) i,j .
We denote by B( 2 ) the Banach space of all infinite matrices representing bounded operators acting on 2 with respect to the standard basis, endowed with the operator norm.We denote also by M ( 2 ) the space of all Schur multipliers with respect to the norm .
By reasons determined by Theorem 8.1 in [4] we will also call this space the space of matriceal measures.
A Toeplitz matrix M is an infinite matrix with the entries m ij = m j−i for all i, j ≥ 1, where (m j ) j is a sequence of complex numbers.Sometimes we use for this matrix the notation M = (m k ) +∞ k=−∞ , and we denote by T the class of all infinite Toeplitz matrices.
Guided by the results in [4] and [10], which characterize the Toeplitz matrices in M ( 2 ) and B( 2) by means of the associated Fourier series we extended in [3] some aspects of the classical Fejer's theory to the case of infinite matrices.We introduced there a space of continuous matrices C( 2) and a space of integrable matrices L 1 ( 2 ) and studied relations between them and the more classical B( 2 ) and M ( 2 ) and also described some of their properties.
An important and well-known fact (see [10]) is the following: B (l 2 ) ∩ T may be identified with L ∞ (T) in the manner indicated in Theorem 0, and we will write it rather abusive as B (l 2 ) ∩ T = L ∞ (T) .Similarly, M (l 2 ) ∩ T =M (T) (see [4]) with equality of norms.Here of course, M (T) is the space of all regular bounded Borel measures μ on T, endowed with the norm where |μ| is the bounded variation of μ.
So, we identify Toeplitz matrices with functions and we intend to extend some classical results from the Toeplitz matrices setting to the more general infinite matrices.
Starting from these remarks, we consider, in the second section a class of matrix spaces similar to Lebesgue spaces L p (T), 1 ≤ p ≤ ∞, denoted by We initiate also the study of the matriceal Hölder inequality.A naive extension of the classical Hölder inequality is, unfortunately, not valid.
We give some special classes of matrices, called matriceal Hardy spaces, such that this inequality will be true for them.
If A = (a ij ) i,j≥1 we will denote by A t = (a ji ) i,j≥1 the transpose of A.
For every infinite matrix A, let us denote by Other notations used in the paper will be introduced as needed.
The following result is well-known: (See [10].) We recall now that A k plays the role of the " k th Fourier coefficient of the matrix A." It is well-known that for each f ∈ L ∞ (T) whose Fourier coefficients are a n , n ∈ Z, we have So, the following definition (see [3]) is natural: . ., for n ∈ N * , the matriceal Fejer sum of the order n associated to A.
Then we call a matrix A to be a continuous matrix and we write A ∈ C( 2 ) if the following relation holds: Obviously C( 2 ) endowed with the operator norm becomes a Banach space.
Theorem 0 allows us to write the formula where by [H] * we denote the image of the space H of matrices by the correspondence A → f A .
Remark 2. For brevity we write in what follows equations like the previous one in the following manner: In the sequel we use another notation, more appropriate to our aims, for the entries of the matrix A, namely we put , where l ∈ N * , be the matrix given by We call the matrix A (l) , the l th -corner matrix associated to A. Now, if for any corner-matrix We can identify the matrix A = (A (l) ) l∈N * with its sequence of associated distributions f not = (f l ) l∈N * , writing this fact as By Theorem 0 we have the following correspondences: Then, of course, it follows Having these notions in mind we introduce a commutative product of infinite matrices: Definition 3. Let A = A f and B = A g two infinite matrices of finite band type.We introduce now the commutative product given by Remark 4. (1) We mention that in the previous definition we took A = A f , B = A g infinite matrices of finite band type since, f and g being trigonometric polynomials, we may consider the product f g.

Lebesgue spaces of infinite matrices and Hölder inequality
We intend now to introduce some spaces of infinite matrices which are similar to classical Lebesgue spaces L p (T), 1 ≤ p ≤ ∞.
We need also a space of infinite matrices which is similar to Hilbert space L 2 (T) and, in fact, extends it.
Let first consider the space where I is the identity operator, 0 otherwise, and B ( 2 , ∞ ) is the space of all matrices representing bounded linear operators from 2 into ∞ , endowed with operator norm. .
Motivated by Definitions 1 and 5 we consider the space It is easy to see that and also L 2 ( 2 ) ∩ T = L 2 (T) .L 2 ( 2 ) can be regarded as a matriceal extension of the familiar Hilbert space L 2 (T).
A good reason for this is the fact that we may introduce in L 2 ( 2 ) an extension of the usual scalar product.
Now we give the analogue of Theorem 6 for L 1 ( 2 ) .
Theorem 7. Let P be the Arveson projection.Then P : Proof.Let A = (a l j ) j∈Z, l≥1 .We have: .
Hence P : M ( 2 ) → M (T) is a bounded projection with P ≤ Λ .By Definition 5 this proves also the assertion concerning L 1 ( 2 ).
It is also easy to see that: Theorem 8.The Arveson projection P, given by a fixed Banach limit Λ, maps continuously L 2 ( 2 ) onto L 2 ( 2 ) ∩ T and P ≤ Λ .
So, by Theorem 6 , it follows that C( 2 ) ∩ T = C(T) is a complemented subspace of C( 2) with respect to the Arveson projection P and by Theorem 8 the space We may consider now the complex interpolation space (see [6], [9]) and call it the matriceal Lebesgue space L p ( 2 ), for 2 ≤ p ≤ ∞.Now we can use a general result of interpolation theory (See [9], 1.17.1,Theorem 1.) Theorem A. Let (A 0 , A 1 ) be an interpolation pair of Banach spaces and B a complemented subspace of A 0 + A 1 , such that the corresponding projection P maps continuously A 0 into A 0 and A 1 into A 1 .Let F be an interpolation functor.Then (A 0 ∩ B, A 1 ∩ B) is an interpolation pair and

Indeed we put in Theorem
p , for 2 ≤ p ≤ ∞.Then, by Theorems 6 and 8, B and P satisfy the conditions of Theorem A, hence (C(T), L 2 (T)) is an interpolation pair and Similarly, using Theorems 7 and 8 in Theorem A, we get Here, of course, we denote In order to justify the introduction of above matrix spaces we would like to prove a version of Hölder inequality in a subspace of L p ( 2 ), for 2 ≤ p ≤ ∞.Definition 9. We denote by H 1 ( 2 ) = {A ∈ L 1 ( 2 ); A upper triangular} and we call H 1 ( 2 ) the matriceal Hardy space of order 1.
In what follows we consider only upper triangular matrices.
Then, by straightforward computation, we get: Remark 10.Let A ∈ T such that f A (t) be an analytic polynomial P n (e it ) and B an upper triangular matrix.Then where A • B means the usual non commutative product of matrices.Now we are interested to find a matriceal version of the well-known Hölder inequality.
At the first glance we are tempted to look at the following naïve matriceal Hölder inequality in case p = 1, q = ∞.Then it follows easily that (A B) Of course there are also other classes of matrices A, B such that A B ∈ L 1 ( 2 ).