Schur multiplier characterization of a class of infinite matrices

Mediterranean Journal of Mathematics, 2019

In this paper, we will consider matrices with entries in the space of operators B(H), where H is a separable Hilbert space, and consider the class of (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vectorvalued functions.

Banach Center Publications, 2010

Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made in [29]. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with the classical Schur multipliers defined by Schur and their characterisation by Grothendieck, and make our way through measurable multipliers studied by Peller and Spronk, operator multipliers defined by Kissin and Shulman and, finally, multidimensional Schur and operator multipliers developed by Juschenko and the authors. We point out connections of the area with Harmonic Analysis and the Theory of Operator Integrals. 1. Classical Schur multipliers For a Hilbert space H, let B(H) be the collection of all bounded linear operators acting on H equipped with its operator norm • op. We denote by ℓ 2 the Hilbert space of all square summable complex sequences. With an operator A ∈ B(ℓ 2), one can associate a matrix (a i,j) i,j∈N by letting a i,j = (Ae j , e i), where {e i } i∈N is the standard orthonormal basis of ℓ 2. The space M ∞ of all matrices obtained in this way is a subspace of the space M N of all complex matrices indexed by N × N. It is easy to see that the correspondence between B(ℓ 2) and M ∞ is one-to-one. Any function ϕ : N × N → C gives rise to a linear transformation S ϕ acting on M N and given by S ϕ ((a i,j) i,j) = (ϕ(i, j)a i,j) i,j. In other words, S ϕ ((a i,j) i,j) is the entry-wise product of the matrices (ϕ(i, j)) i,j and (a i,j) i,j , often called Schur product. The function ϕ is called a Schur multiplier if S ϕ leaves the subspace M ∞ invariant. We denote by S(N, N) the set of all Schur multipliers. Let ϕ be a Schur multiplier. Then the correspondence between B(ℓ 2) and M ∞ gives rise to a mapping (which we denote in the same way) on B(ℓ 2). We first note that S ϕ is necessarily bounded in the operator norm. This follows from the Closed Graph Theorem; indeed, suppose that A k → 0 and S ϕ (A k) → B in the operator norm, for some elements A k , B ∈ B(ℓ 2),

Proceedings of the American Mathematical Society, 2010

We compute the norm of the restriction of a Schur multiplier, arising from a multiplication operator, to a coordinate subspace. This result is used to generalize Wielandt's minimax inequality. Furthermore, we compute various s-numbers of an elementary Schur multiplier and determine criteria for membership of such multipliers in certain operator ideals.

2020

We construct a family of Schur multipliers for lower triangular matrices on ℓ^p, 1<p<∞ related to θ-summability kernels, a class of kernels including the classical Fejer, Riesz and Bochner kernels. From this simple fact we derive diverse applications. Firstly we find a new class of Schur multipliers for Hankel operators on ℓ^2, generalizing a result of E. Ricard. Secondly we prove that any space of analytic functions in the unit disc which can be identified with a weighted ℓ^p space, has the property that the space of its multipliers is contained in the space of symbols g that induce a bounded generalized Cesáro operator T_g.

Proceedings of the American Mathematical Society, 1982

Let f f be a continuously differentiable function on [ − 1 , 1 ] [ - 1,1] satisfying | f ′ ( t ) | ⩽ C | t | α \left | {f’(t)} \right | \leqslant C{\left | t \right |^\alpha } for some 0 > C 0 > C , α > ∞ \alpha > \infty and all − 1 ⩽ t ⩽ 1 - 1 \leqslant t \leqslant 1 , and let λ = ( λ i ) ∈ l r \lambda = ({\lambda _i}) \in {l_r} satisfy − 1 ⩽ λ i ⩽ 1 - 1 \leqslant {\lambda _i} \leqslant 1 for all i i . Then \[ a f , λ = ( f ( λ i ) − f ( λ j ) λ i − λ j ) {a_{f,\lambda }} = \left ( {\frac {{f({\lambda _i}) - f({\lambda _j})}} {{{\lambda _i} - {\lambda _j}}}} \right ) \] is a Schur-Hadamard multiplier from C p 1 {C_{{p_1}}} into C p 2 {C_{{p_2}}} for all p 1 {p_1} , p 2 {p_2} satisfying 1 ⩽ p 2 ⩽ 2 ⩽ p 1 ⩽ ∞ 1 \leqslant {p_2} \leqslant 2 \leqslant {p_1} \leqslant \infty and p 2 − 1 ⩽ p 1 − 1 + α / r p_2^{ - 1} \leqslant p_1^{ - 1} + \alpha /r .

Applied Mathematics and Computation, 2009

It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246-251 [2]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009-1030] [14], that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant, as well as its diagonal-Schur complement. Also, if a matrix is an H-matrix, then its Schur complement and diagonal-Schur complement are H-matrices, too, see [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Alg. Appl. 389 (2004) 365-380] [13]. Recent research, see [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Alg. Appl. 378 (2004) 231-244 [12]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009-1030] [14], showed that the similar statements hold for some special subclasses of H-matrices. The aim of this paper is to give more invariance results of this type, and simplified proofs for some already known results, by using scaling approach.

2017

Dans le premier chapitre, nous commencons par definir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques proprietes des classes de Schatten. La fin du chapitre est dediee a l’etude des espaces de Bochner a valeurs dans l'espace des operateurs factorisables par un espace de Hilbert. Le deuxieme chapitre est consacre aux multiplicateurs de Schur lineaires. Nous caracterisons les multiplicateurs bornes sur B(Lp, Lq) lorsque p est inferieur a q puis appliquons ce resultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisieme chapitre, nous caracterisons, au moyen de multiplicateurs de Schur lineaires, les multiplicateurs de Schur bilineaires continus a valeurs dans l'espace des operateurs a trace. Dans le quatrieme chapitre, nous donnons divers resultats concernant les operateurs integraux multiples. En particulier, nous caracterisons les operateurs integraux triples a valeurs dans l'espace d...

Bulletin of the Australian Mathematical Society, 2008

The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying f (S) ⊆ S for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f (A)) = Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

Mathematische Annalen, 2002

We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class S p for 0 < p < 1. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of S p. We also give a characterization of the Hankel-Schur multipliers of S p whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of S p. Finally, we return to Hankel-Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle.

Proceedings of the American Mathematical Society, 2008

We give a formula for Markov dilation in the sense of Anantharaman-Delaroche for real positive Schur multipliers on B(H).