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2019, Mediterranean Journal of Mathematics
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),
Taiwanese Journal of Mathematics, 2018
In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces B(H) which define bounded operators acting on ℓ 2 (H) and use it to get the description of the right Schur multipliers acting on ℓ 2 (H) in terms of certain operator-valued measures.
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.
Quaestiones Mathematicae, 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 matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in B(H). We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as H ∞ (D) and A(D) when dealing with upper triangular matrices.
2007
We introduce multidimensional Schur multipliers and characterise them generalising well known results by Grothendieck and Peller. We define a multidimensional version of the two dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C*-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding C*-algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corres...
Banach Journal of Mathematical Analysis, 2016
Let E i be Banach spaces and let X E i be Banach spaces continuously contained in the spaces of E i-valued sequences (x(j)) j ∈ E N i , for i = 1, 2, 3. Given a bounded bilinear map B : E 1 × E 2 → E 3 , we define (X E 2 , X E 3) B the space of B-multipliers between X E 2 and X E 3 to be the set of sequences (λ j) j ∈ E N 1 such that (B(λ j ,x(j))) j ∈ X E 3 for all (x(j)) j ∈ X E 2 and the Hadamard projective tensor product X E 1 B X E 2 , consisting of those elements in E N 3 that can be represented as n j B(x n (j),ŷ n (j)), where (x n) n ∈ X E 1 , (y n) n ∈ X E 2 and n x n X E 1 y n X E 2 < ∞. We will analyze some properties of these two spaces, relate them and compute the Hadamard tensor products and the spaces of vectorvalued multipliers in several cases, getting applications in the particular case E = L(E 1 , E 2) and B(T, x) = T (x).
Proceedings of the Indian Academy of Sciences - Section A, 2006
In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.
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).
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 < q ≤ p < +∞, 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'(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).
Mathematical Transactions of the Academy of Sciences of the Lithuanian SSR, 1973
HILBERT SPACES OF OPERATOR-VALUED FUNCTIONS E. Senkene and A, Tempel'man UDC 519,21 Hilbert spaces of scalar functions with reproducing kernels were introduced and studied in [1]. In [3] and [6] the properties and probability-theoretical applications of Hilbert spaces of vector-valued functions with operator-valued reproducing kernels were considered. The present paper is devoted to the study of Hilbert spaces of operator-valued functions with reproduction kernels; a number of facts which are well known in the scalar case (cf. [1]) are extended to such spaces. Let :~C be a separable Hilbert space; < .,. > :)~ and !!'ll~ denote the scalar product and norm in ~E; ~C is the Banaeh ring of all nuclear operators in }~ (with norm I[A ]]c}~=tr [(AA*)]~); • is the Banaeh algebra of linear operators (with norm It A tip = sup [I Ax [I iC); T is an arbitrary set; H is the Hilbert space con-. , ~II:E= i ' sisting of ~,C valued functions defined on T; <., * >H and I1 ' tl H denote the scalar product and norm in H. Definition 1. A ~C-valued function R(s, t), s, t, tE T, is called an operator-valued reproducing kernel of the space H if for any function A(.)~H and for any Be2 and s~ T a) the function BR(s, .)EH; b) <A(.), BR(s, ")>H = tr[A(s)B*], where tr[A(s)B*] denotes the trace of the operator A(s)B* (the "reproducing" property). We note that if B *= x | y is a one-dimensional operator defined by the formula (x| < z, y> }~x, then tr [A (s)B*]= < A (s)x, y> ??; thus, on the basis of the properties of the function A(.) as an element of the space H we can reconstruct its value at each point of s ~ T; in particular, if {e 1, e 2, e s .... } is a basis of }~, then a:~ (s)=~r [A (~) (e~| < 4 (s) e~, ej> }~ are the matrix elements of the operator A(s) with respect to this basis.
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 ,
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.
Czechoslovak Mathematical Journal, 2010
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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.
Integral Equations and Operator Theory, 2010
We estimate the completely bounded norms, the completely p-nuclear norms, and the completely p-summing norms of certain multiplication operators and Schur multipliers.
Linear Algebra and its Applications, 2009
AMS classification: 46B70 46E40 47A30 47A55 47A63 47B10 47B47 47B49 47D50 Keywords: Schatten ideals Norm of elementary operator Norm reconvexization
Journal of Functional Analysis, 2001
A multilinear version of Schur's test is obtained for products of L p spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.
2019
By L(X,Y ), we denote the space of all continuous linear operators between the normed spaces X and Y . In [15], Swartz introduced the (bounded) multiplier space for the series ∑ Tj as: M∞( ∑ Tj) = {x = (xj) ∈ `∞(X)| ∑ j Tjxj converges}, where (Tj) ⊆ L(X,Y ). Recently in [6], Altay and Kama defined the vector valued multiplier space M∞ C (T ) of Cesàro convergence by using Cesàro summability method as follow: M∞ C (T ) = {x = (xk) ∈ `∞(X)| ∑ k Tkxk is Cesàro convergent}. In this paper, we introduce the vector valued multiplier spaces SΛ(T ) and SwΛ(T ) by means of Λ− convergence and a sequence of continuous linear operators and study a series of some properties of these spaces.
A locally integrable function m(ξ, η) defined on R n × R n is said to be a bilinear multiplier on R n of type (p 1 , p 2 , p 3) if Bm(f, g)(x) = Z R n Z R nf (ξ)ĝ(η)m(ξ, η)e 2πi(ξ+η,x dξdη defines a bounded bilinear operator from L p 1 (R n) × L p 2 (R n) to L p 3 (R n). The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where m(ξ, η) = M (ξ − η) for a given M defined on R n is also addressed. R n f (x)e −2πi x,ξ dx. We shall use the notation M p,q (R n) (respect.M p,q (R n)), for 1 ≤ p, q ≤ ∞, for the space of distributions u ∈ S ′ (R n) such that u * φ ∈ L q (R n) for all φ ∈ L p (R n) (respect. for the space of bounded functions m such that T m defines a bounded operator from L p (R n) to L q (R n) where T m (φ)(ξ) = m(ξ)f (ξ).) We endow the spaceM p,q (R n) with the "norm" of the operator T m , that is m p,q = T m. Let us start off by mentioning some well known properties of the space of linear multipliers (see [1, 14]): M p,q (R n) = {0} whenever q < p, M p,q (R n) = M q ′ ,p ′ (R n) for 1 < p ≤ q < ∞ and for 1 ≤ p ≤ 2, M 1,1 (R n) ⊂ M p,p (R n) ⊂ M 2,2 (R n). We also have the identificationsM
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...
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