On the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups

We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore, we show that “non-triangular” sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of G is a local Ditkin set.     

Some conditions for hypervaluability of a ring by a totally ordered semigroup

The purpose of this paper is to give some conditions for a ring to be hyperbaluable by a totally ordered semigroup. In § 1 we give some preliminary definitions and results, and a bibliographical note. The theorem in § 2 provides a sufficient condition for hypervaluability. The theorem in § 3 provides a necessary and sufficient condition for hypervaluability. In § 4 we provide an example of a ring satisfying the conditions of theorem 2 and is therefore hypervaluable by a totally ordered semigroup.     

Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

L-Fourier multipliers for the Dunkl operator on the real line

We consider Fourier multipliers for L^p associated with the Dunkl operator on and establish a version of Hörmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials.     

Fourier transforms of Schwartz functions on Chébli-Trimèche hypergroups

The Fourier transform for Schwartz spaces on Chébli-Trimèche hypergroups is studied, leading to results on approximation to the identity for functions and distributions on the half-line. In particular it is shown that the heat and Poisson kernels on the half-line form approximate units in various function spaces. A characterization of the convolution of a tempered distribution and a Schwartz function is also given.     

Induced representations and hypergroup homomorphisms

The purpose of this paper is to continue the study of induced representations of hypergroups. We introduce the concept of Weil subhypergroups and Weil homomorphisms and derive a Weil formula for kernels of Weil homomorphisms. Making use of this decomposition formula, we prove in the main theorem that lifting a representation via a Weil homomorphism is compatible with the inducing process. As a further consequence we obtain that a character of a supernormal subhypergroup is inducible, if and only if it admits an extension, while in general the extensibility of a character does not imply its inducibility.This research project is supported by the DFG.     

Spectral synthesis on multivariate polynomial hypergroups

In a recent paper [16] we presented some results concerning spectral analysis and spectral synthesis on polynomial hypergroups in a single variable. Now we show that using Hilbert’s Nullstellensatz, the Noether-Lasker-theorem and the Ehrenpreis-Palamodov-theorem the ideas of [16] can be extended to prove spectral analysis and spectral synthesis on any multivariate polynomial hypergroup. Author’s address: Institute of Mathematics, University of Debrecen, Egyetem tér 1, 4032 Debrecen, Hungary     

Locally dense finite lattice packings of spheres

We consider finite packings of unit-balls in Euclidean 3-spaceE ³ where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL ³E³. In particular we show that the facets ofP induced by densest sublattices ofL ³ are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.The results support, to some extent, the hypothesis that real crystals grow preferably such that they need little volume, i.e that they are locally dense.Dedicated to A. Florian on the occasion of this 60th birthday     

Eigenvalues of integral operators onL 2(I) given by analytic kernels

Let {_n} _n=0 be the eigenvalue sequence of a symmetric Hilbert-Schmidt operator onL ₂(I). WhenI is an open interval, a necessary condition for {_n} _n=0 to be in the sequence space is obtained. WhenI is a closed bounded interval, sufficient conditions for {_n} _n=0 to be in the sequence space ^– are obtained.     

Krengel-Lin decomposition for probability measures on hypergroups

A Markov operator P on a σ-finite measure space (X,Σ,m) with invariant measure m is said to have Krengel-Lin decomposition if L²(X)=E₀⊕L²(X,Σ_d) where E₀={f∈L²(X)∣‖Pⁿ(f)‖→0} and Σ_d is the deterministic σ-field of P. We consider convolution operators and we show that a measure λ on a hypergroup has Krengel-Lin decomposition if and only if the sequence converges to an idempotent or λ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups.     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

Dual analytic models of seminormal operators

In [6] (after Clancey's work [2]), Martin and Putinar introduced their two-dimensional functional model of a hyponormal operator, which reduces it to the multiplication by the independent variable in a space of distributions. Here we define another model which does (almost) the same for the adjoint operator. We also explain a close relation between these two models and dual bundle shift models of linear operators introduced in [13]. As application, an estimate of the effectual rational multiplicity of hyponormal operators is given.The research described in this publication was made possible in part by Grant No. NW8000 from the International Science Foundation     

Some remarks on the construction of quantum symmetric spaces

We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of q-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in U _q(g(n, )) and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter.The author acknowledges financial support by the Japan Society for the Promotion of Science (JSPS) and the Netherlands Organization for Scientific Research (NWO).     

Hardy and Cowling–Price theorems associated with the Jacobi–Cherednik operator

This paper is devoted to the proof of Hardy and Cowling–Price type theorems for the Fourier transform tied to the Jacobi–Cherednik operator.     

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω$\Omega$ of positive definite real matrices. The highest weight representations of the group Sp(n,R)$\mathrm{Sp}(n,\mathbb{R})$ play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)$\Omega +i\mathrm{Sym}(n,\mathbb{R})$. We then use the Laplace transform to carry the Lie algebra action over to L²(Ω,d_μν)$L^2(\Omega ,\mathrm{d}{\mu }_{\nu })$. The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).$\mathfrak{sl}(2,\mathbb{R}).$     

Some remarks on a paper by Liu and van Rooij

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.     

Bargaining outcomes in patent licensing: Asymptotic results in a general Cournot market

We study asymptotic bargaining outcomes in licensing a patented technology of an external patent holder to firms in a general Cournot market. Our results are as follows: when the number of firms is large, the bargaining set for each permissible coalition structure suggests that the patent holder should extract the entire profits of all licensees. The outcome that the bargaining finally reaches exactly coincides with the non-cooperative outcome, and it cannot be improved upon even by any objections with almost zero cost. Thus, it is strongly stable. The fair allocation represented by the Aumann-Drèze value is, however, not realized as such a stable bargaining outcome.