Integrable factors in compact Schur multipliers

It is shown that a Schur multiplier is compact if and only if it is the Schur product of two multipliers, one of which is a Hankel-Schur multiplier generated by an integrable function. This is illuminated by factoring exotic, singular measures and is brought into relation with Paley set-based multipliers.

     

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allow to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels and positive definite functions in this setting and we show how they can be obtained using the extension operator and the slice hyperholomorphic product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges–Rovnyak space.     

Block-matrix generalizations of infinite-dimensional schur products and schur multipliers

We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.     

Noncommutative Figà-Talamanca–Herz Algebras for Schur Multipliers

In this work, we introduce a noncommutative analogue of the Figà-Talamanca–Herz algebra A _p (G) on the natural predual of the operator space \frakM_p,cb{\frak{M}_{p,cb}} of completely bounded Schur multipliers on the Schatten space S _p . We determine the isometric Schur multipliers and prove that the space \frakM_p{\frak{M}_{p}} of bounded Schur multipliers on the Schatten space S _p is the closure in the weak operator topology of the span of isometric multipliers.     

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.     

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.     

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).     

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.

     

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.      

Un théorème de type Beurling–Lax dans la boule unitéA Beurling–Lax type theorem in the unit ball

We prove a Beurling–Lax theorem for a family of reproducing kernel Hilbert spaces of functions analytic in an open subset of the unit ball containing the origin. The spaces under consideration are characterized by functions called Schur multipliers. Using the theory of linear relations in Pontryagin spaces we also give coisometric realizations of Schur multipliers. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 349–354.     

Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces

We establish Jackson-type and Bernstein-type inequalities for multipliers on Herz-type Hardy spaces. These inequalities can be applied to some important operators in Fourier analysis, such as the Bochner-Riesz multiplier over the critical index, the generalized Bochner-Riesz mean and the generalized Able-Poisson operator. This work was supported by Key Academic Discipline of Zhejiang Province of China and National Natural Science Foundation of China (Grant Nos. 10571014, 10631080, 10671019)     

Analyse de Fourier,multiplicateurs de Schur sur Sp et ensembles Λ(p)cb non commutatifs

This work deals with various questions concerning Fourier multipliers on L^p, Schur multipliers on the Schatten class S^p as well as their completely bounded versions when L^p and S^p are viewed as operator spaces. We use for this aim subsets ofenjoying the Λ(p)_cb-property which is much stronger than the usual Λ(p)-property. We start by studying the notion of Λ(p)_cb-sets in the general case of an arbitrary discrete group before turning to .