Certain Schur-Hadamard multipliers in the spaces $C\sb{p}$

Czechoslovak Mathematical Journal, 2010

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

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, 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.

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.

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),

2017

Let (Ω_1, F_1, μ_1) and (Ω_2, F_2, μ_2) be two measure spaces and let 1 ≤ p,q ≤ +∞. We give a definition of Schur multipliers on B(L^p(Ω_1), L^q(Ω_2)) which extends the definition of classical Schur multipliers on B(ℓ_p,ℓ_q). Our main result is a characterization of Schur multipliers in the case 1≤ q ≤ p ≤ +∞. When 1 &lt; q ≤ p &lt; +∞, ϕ∈ L^∞(Ω_1 ×Ω_2) is a Schur multiplier on B(L^p(Ω_1), L^q(Ω_2)) if and only if there are a measure space (a probability space when p≠ q) (Ω,μ), a∈ L^∞(μ_1, L^p(μ)) and b∈ L^∞(μ_2, L^q&#39;(μ)) such that, for almost every (s,t) ∈Ω_1 ×Ω_2, ϕ(s,t)=〈 a(s), b(t) 〉. Here, L^∞(μ_1, L^r(μ)) denotes the Bochner space on Ω_1 valued in L^r(μ). This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on B(ℓ_p,ℓ_q).

Journal of Soviet Mathematics, 1985

In this paper there is given a sufficient condition for a Hankel matrix F F to belong to the space of Schur multipliers of all bounded operators in 12 (or, what is the same, to the tensor algebra V2). It is shown that if ~ is a nonnegative function on ~ , such that 4/~, {o~Jj~ I is a sequence of integers, ~/~I ,

International Mathematics Research Notices, 2018

Journal of Mathematical Inequalities, 2007

The Schur-convexity on the upper and the lower limit of the integral for a mean of the convex function is researched. As applications, a generalized logarithmic mean with a parameter is obtained and a relevant double inequality that is a extension of the known inequality is established.