Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Erich Toeplitz: Die Menorah des Chanukkafestes [Rezension]

N. H.

Asymptotically Toeplitz Hessenberg Operators and the Rieman mapping

In a recent work the authors have established a relation between the limits of the elements of the diagonals of the Hessenberg matrix D associated with a regular measure, whenever those limits exist, and the coe?cients of the Laurent series expansion of the Riemann mapping ?(z) of the support supp(?), when this is a Jordan arc or a connected nite union of Jordan arcs in the complex plane C. We extend here this result using asymptotic Toeplitz operator properties of the Hessenberg matriz.

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Tau Functions and the Limit of Block Toeplitz Determinants

International audience

A rotation method which gives linear Lp-Estimates for powers of the Ahlfors-Beurling operator

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Recomanacions sobre pràctiques higièniques per embotits fermentats tradicionals

La seguretat i qualitat dels aliments és actualment una de les majors preocupacions dels consumidors europeus i una prioritat política en Estats Units i Europa, a través del Programa Agenda 2000 i el Llibre Blanc de seguretat alimentària (2000). Aquest llibre blanc va indicar 3 prioritats: 1. Crear una Agència Europea dels Aliments, 2. Implementar un enfocament coherent de la granja a la taula en termes de legislació alimentària, 3. Establir el principi a partir del qual la responsabilitat clau de la seguretat alimentària recau en els operadors alimentaris. Els estats membres són els encarregats de vigilar i controlar a aquests productors i la Comissió Europea comprova l’eficiència dels Estats per a controlar-los (subsidiari). La legislació comunitària estableix els objectius de les normes d’higiene (en vigor des del 20 de maig del 2004) a través de l’ús d’un enfocament integrat de tota la cadena alimentària (des de la granja a la taula), per a garantir un alt nivell de protecció en termes de seguretat sanitària dels productes alimentosos. A Europa, els productes artesanals i/o tradicionals gaudeixen d’una posició privilegiada als ulls dels consumidors que demanden productes autèntics, de qualitat i d’origen garantit.

Variational inequalities in Hilbert spaces with measures and optimal stopping problems

We study the existence theory for parabolic variational inequalities in weighted L2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coeficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

Net spaces and boundedness of integral operators

In this paper we introduce new functional spaces which we call the net spaces. Using their properties, the necessary and sufficient conditions for the integral operators to be of strong or weak-type are obtained. The estimates of the norm of the convolution operator in weighted Lebesgue spaces are presented.

Generalized multiresolution analyses with given multiplicity functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L2(Rn), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

Analytic approximation of matrix functions in Lp

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Differentiability of functions of contractions

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Known and new results on absolute convergence of Fourier integrals

New sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension d &= 2.

Sharp A₂ inequality for haar shift operators

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Weak-type estimates for Haar shift operators: Sharp power on the A(p) characteristic

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