Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . 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. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

On Components of the Partial Schur Multiplier

In this paper we study properties of the partial Schur multiplier. As an application a characterization of components of the partial Schur multiplier for some groups is given. We also describe the classical Schur multiplier of elementary abelian 2-groups by using partial factor sets.     

On dimension and homological methods of the (higher) Schur multiplier of a pair of Lie algebras

In this article, some inequalities of the dimension of Schur multiplier of pairs of Lie algebras will be obtained and new upper bound will be compared with the existing ones in the literature. Furthermore, by using homological methods, we will derive some properties of the higher Schur multiplier of a pair of Lie algebras and give some isomorphisms that generalize some known results of Stallings in group theory setting.     

Updating the Multipliers Associated with Inequality Constraints in an Augmented Lagrangian Multiplier Method

This paper contributes to the development of the field of augmented Lagrangian multiplier methods for general nonlinear programming by introducing a new update for the multipliers corresponding to inequality constraints. The update maintains naturally the nonnegativity of the multipliers without the need for a positive-orthant projection, as a result of the verification of the first-order necessary conditions for the minimization of a modified augmented Lagrangian penalty function.In the new multiplier method, the roles of the multipliers are interchanged: the multipliers corresponding to the inequality constraints are updated explicitly, whereas the multipliers corresponding to the equality constraints are approximated implicitly. It is shown that the basic properties of local convergence of the traditional multiplier method are valid also for the proposed method.     

On the norm of an idempotent Schur multiplier on the Schatten class

We show that if the norm of an idempotent Schur multiplier on the Schatten class lies sufficiently close to , then it is necessarily equal to . We also give a simple characterization of those idempotent Schur multipliers on whose norm is .

     

Semigroups of Herz–Schur multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz–Schur multipliers generated by a proper function (see Theorem 1.2). It is then shown that a (not necessarily proper) generator of a semigroup of Herz–Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz–Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function (see Theorem 1.6).     

On the dimension of the Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.     

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

     

Bounds on the order of the Schur multiplier of p-groups

In 1956, Green provided a bound on the order of the Schur multiplier of p-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other inputs to the function, such as, the minimal number of generators of the group and the order of the derived subgroup. We strengthen these bounds by adding another input, the group's nilpotency class. The specific cases of nilpotency class 2 and maximal class are discussed in greater detail.     

The Schur Multiplier of a Pair of Groups

In this article we develop the theory of a Schur multiplier for pairs of groups. The idea of such a multiplier is implicit in the work of J.-L. Loday (1978) and others on algebraicK -theory, and in the work of Eckmann et al. (1972) and others on group homology. In contrast to their work, we focus on the general group-theoretic properties of the multiplier. These properties are systematically derived from: 1) the functoriality of the multiplier; 2) an exact homology sequence; 3) and a transfer homomorphism.     

Wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

ON A CONNECTION BETWEEN THE SCHUR MULTIPLIER AND THE BRAUER GROUP OF A FIELD

ABSTRACT

Let k be a field and suppose ζ_n is a primitive n-th root of unity contained in k. We denote by Sr(k) = H² (G_k,C^x) the Schur multiplier of k. We observe: If the n-part of Sr(k) is trivial, then every element of order n in the Brauer group Br(k) can be represented as a cyclic algebra of the form

(E,σζ_n)

with the same ζn throughout. We apply this in particular to the case of a global field k and conclude by discussing the Schur multipliers of certain other types of fields.     

A Hörmander type multiplier theorem for multilinear operators

In this paper, we prove a Hörmander type multiplier theorem for multilinear operators. As a corollary, we can weaken the regularity assumption for multilinear Fourier multipliers to assure the boundedness.     

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

Given a collection of test functions, one defines the associated Schur–Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur–Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with ${{\mathbb C}^{N}}$ of the scalar-valued version.     

Some Inequalities for the Order of the Schur Multiplier of a Pair of Groups

Let (H, N) be a pair of finite p-groups, in which N is a normal subgroup of H. In the present article, some upper bounds for the order of the commutator subgroup [N, H] are given. It is also constructed an upper bound for the order of the Schur multiplier of the pair of finite p-groups.     

Some 3-Generator, 3-Relation Finite 2-Groups

Using computational methods, we first show that there are exactly eighteen 3-generator 2-groups of order 2¹⁰ with trivial Schur multiplier all having deficiency zero. We next generalize one of the groups obtained to exhibit two infinite classes of 3-generator, 3-relation finite 2-groups of high nilpotency class providing an affirmative answer to a problem posed by Havas et al.