Functional Analysis and Applications Seminar 

We will essentially discuss two points in the connection with extremal functions of kernels of Toeplitz operators on Hardy spaces. The first one concerns divisor properties of such extremal functions. It turns out that in many situations such a division has nice properties like being a contraction (case of Hedenmalm's canonical divisors in the Bergman space), or even an isometry (inner functions in the Hardy space). Concerning extremal functions of kernels of a Toeplitz operator, the question has been considered in the larger class of nearly invariant subspaces by Hitt. He proved that in the Hilbert space situation $H^2$, the division by the extremal function of a nearly invariant subspace is isometric. The situation changes drastically even for Toeplitz kernels when one switches to the non Hilbert case ( $p \neq 2$), where, depending on the parameter $p \gt 1$, one can in general only expect a control on the division or on the multiplication by the extremal function. Examples show that two-sided estimates cannot be expected in general. The second part of the talk will be devoted to the investigation of invertibility properties of Toeplitz operators by means of the extremal function. This understands that the Toeplitz operator is supposed non injective in order that such an extremal function exists. In this part we have to assume the Hilbert situation $p = 2$. It turns out that certain parameters associated with the extremal function, and that have previously been used by Hayashi to distinguish kernels of Toeplitz operators from general nearly invariant subspaces, enable us to characterize the surjectivity of a (non injective) Toeplitz operator.

The Gelfand-Naimark theorem establishes that any commutative $C^{*}$-algebra $A$ is isometrically $*$-isomorphic to a certain space of continuous functions over a locally compact topological space. In a way, the $C^{*}$-algebra $A$ might be seen as a translation of the topology of the space. It is the aim of this talk to give an overview of how the generalization of these ideas to non-commutative $C^{*}$-algebras has lead to a non-commutative notion of topology. A résumé of some recent results and of on going research will be presented.

In my talk I will consider the recent developments in the study of global solvability of nonlinear equations. We will consider a quite large class of nonlinear mappings generated by the nonlinear elliptic problems related to the nonlinear pseudodifferential opearators. Here the geometrical properties of corresponding mappings will play a crucial role, which will allow us to define topological and other invariants. We will especially focus an application of these new ideas to the global solvability of nonlinear Riemann-Hilbert problems for general domains.

The problem of heat conduction of 2D bounded composite material with symmetrically situated inclusions having different conductivity is reduced to a vector-matrix Riemann boundary value problem with a piecewise constant matrix. The proposed method generalizes an approach by V.V. Mityushev presented in the book by V.V. Mityushev and S.V. Rogosin on Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions , CRC Press, 1999. The talk is based upon joint work with Sergei Rogosin.

The melting problem with several dendrits is reduced to a free boundary Hele-Shaw problem for a multiply connected domain. Its local in time solvability is studied on the base of a variant of the Nirenberg-Nishida theorem concerning the Cauchy-Kovalevsky problem and functional equations in complex domains. Other approaches to the above problems are discussed as well. The talk is based upon joint work with Tatsjana Vaiteakhovich.

It was shown in [1] that if a bounded analytic solution to the Riemann-Hilbert problem $G h_+ = h_-$ is known which consists of corona pairs, then $G$ admits a canonical generalized factorization, i.e., the Toeplitz operator with symbol $G$ is invertible. But what if the solution does not satisfy the corona conditions? And, to begin at the beginning, how to get a particular bounded analytic solution to the Riemann-Hilbert problem? We adress these questions by studying a class of triangular matrix symbols which illustrates the problems involved in those questions and for which we can find answers.

1. Bastos, M. A., Karlovich, Y. I. and dos Santos, A. F., Oscillatory Riemann-Hilbert problems and corona theorem, Journal of Functional Analysis 197 (2003) 347-397.

Problemas de difracção de ondas têm sido estudados em cooperação com investigadores europeus e americanos no âmbito de JNICT/BMFT project 423/1(94-98) e project 423/2 (95-99), FCT/FEDER/POCTI/MAT/34222 (99) e 59972 (04). Dois dos tópicos de investigação são a difracção por cunhas e por redes periódicas. A apresentação desta investigação no âmbito da Feira do Conhecimento e da Inovação (Abril 2006) foi integrada na iniciativa da divulgação de projectos dum centro de Matemática, o CEMAT, junto dum público heterógeneo. Nesta apresentação mostraremos o multimédia que preparámos para a ocasiâo e tentaremos analisar a importãncia e a projecção desta iniciativa pioneira. Procuraremos discutir a relevância dos tópicos de investigação escolhidos na área dos Problemas de Difracção e da Teoria de Operadores.

This talk deals with joint work with W.N. Everritt. The link of the sampling/interpolation theorem of Shannon-Whittaker with the original Kramer sampling theorem is considered. Also, the connection of these two significant results with boundary value problems associated with linear ordinary differential equations as defined on intervals of real line is specified. The results given in this talk are concerned with the generation from first-order linear, ordinary boundary value problems of Kramer analytic kernels which introduce analytic dependence of the kernel on the sampling parameter. These kernels are represented by unbounded self-adjoint differential operators in Hilbert function spaces. Necessary and sufficient conditions are given to ensure that these differential operators have a simple, discrete spectrum which then allows the introduction of the associated Kramer analytic kernels. Finally, the corresponding analytic interpolation functions are defined with the required properties, to give the Lagrange interpolation series.

The Fredholm properties of Toeplitz operators with \(2\times 2\) matrix symbols \(G\), which are essentially bounded on the real line, are studied in connection with some properties of a solution to a homogeneous linear Riemann-Hilbert problem with coefficient \(G\). Conditions for the Toeplitz operator to be Fredholm are obtained and, if that is the case, formulas for the factors in a generalized factorization of \(G\) are given in terms of the solutions to a pair of non-standard corona problems. These results are used to establish partial index estimates for Daniele-Khrapkov matrix functions.

In the first part of the talk we introduce the Boltzmann equation, discuss its properties and give an overview on existence and uniqueness of solutions. Especially our new results on mapping properties of the Boltzmann collision operator will be presented. Then the Direct Simulation Monte Carlo method (DSMC) which is widely applied in numerics will be explained. In the third part of the talk we present the Stochastic Weighted Particle Method (SWPM) which was introduced in 90's by Rjasanow and Wagner. We apply this method to the numerical solution of the spatially two-dimensional Boltzmann equation. The numerical solution of the Boltzmann equation using naive deterministic methods leads to the amount of numerical work of the order $O(n^8)$, where $n$ denotes the number of discrete velocities in one direction. In the next part of the talk we give an overview on deterministic numerical methods applied to the Boltzmann equation by a number of authors. Then, in the final part of the talk, we present the results of our numerical experiments obtained by a new deterministic approximation of the Boltzmann equation using Fast Fourier Transform.

We explore the extent to which basic differential operators (such as Laplace-Beltrami, Lamé, Navier-Stokes, etc.) and boundary value problems on a hypersurface in ${ℝ}^n$ can be expressed globally, in terms of the standard spatial coordinates in ${ℝ}^n$. The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations. In particular, we obtain explicit representations of the surface deformation tensor, the Laplace-Beltrami operator, the Lamé-Beltrami operator, the operator of isotropic elasticity and the Stokes system on the hypersurface. The obtained representations are helpful in proving existence of fundamental solutions on the manifold and in writing explicit Green formulae for the classical boundary value problems on an open subsurface. Combined with the potential method, the obtained results allow to prove the unique solvability of the aforementioned boundary value problems by a standard scheme. The lecture is based upon joint research with with Dorina Mitrea and Marius Mitrea.

We discuss new higher order asymptotic formulas for traces of Toeplitz matrices with symbols in Hölder-Zygmund spaces. Remainders in these formulas go to zero with the speed depending on the smoothness parameter of the space. The Wiener-Hopf factorization of symbols within Hölder-Zygmund spaces plays an essential role in the proof. These results refine Widom's asymptotic trace formulas and complement Böttcher-Silbermann's higher order asymptotic formulas for determinants of Toeplitz matrices.

The $C$-linear conjugation problem with a piece-wise constant matrix $G(t)$ given on the real axis and discontinuous at the finite numbers of the points $W$ is stated as follows. To find a vector-function $F(z)$ analytic in the upper and lower half planes continuous in the closures of the half planes except the set $W$ where $F(t)$ is bounded. The upper and lower limit values of $F(t)$ on the real axis are linearly related via $G(t)$. A polynomial growth of $F(z)$ at infinity is admitted. This problem is solved in closed form by application of the functional equations method under some restrictions on the dimension of $F(z)$ and $\#W$.

We prove asymptotic formulas for block Toeplitz matrices with symbols admitting right and left Wiener-Hopf factorization such that all partial indices are equal to some integer number. We consider symbols and Wiener-Hopf factorizations in Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Our results complement known formulas for Hoelder continuous symbols due to Boettcher and Silbermann.

Applying a weighted analogue of the Carleson-Hunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators with symbols that are bounded measurable functions with respect to the spatial variable and functions of bounded variation with respect to the dual variable. Replacement of absolutely continuous functions of bounded variation by arbitrary functions of bounded variation allows us to study essentially more general classes of pseudodifferential operators. A symbol calculus and a Fredholm theory for new classes of pseudodifferential operators with non-regular symbols are constructed. In particular, we study pseudodifferential operators with symbols admitting discontinuities of first kind with respect to spatial and dual variables that generate non-commutative algebras of Fredholm symbols.

We prove unique existence of solution for a class of plane wave diffraction problems by a strip with first and second kind boundary conditions. This is done in a Bessel potential spaces framework, and for a real (non-complex) wave number. At the end, results about the regularity (and data dependence) of the solution are exhibited upon the initial setting and the boundary parameters.

Let $G$ be a $2\times 2$ matrix function of Daniele-Khrapkov type. An equivalence between linear Riemann-Hilbert problems with coefficient $G$ and a class of scalar boundary value problems relative to a contour in a Riemann surface $\Sigma$ is established. By studying the solutions of these problems, it can be shown that the solution of the former Riemann-Hilbert problems must satisfy certain relations. In particular, if $G$ admits a canonical bounded factorization, it follows that the factors must have a certain structure.

The local trajectory method is an invertibility method for operator algebras with generators associated to unitary representations of groups. In 1991 Yuri Karlovich proposed a generalization of this method. We present a new generalization of the method and discuss some possible applications.

Since Torsten Ehrhardt succeeded in creating a factorization theory for Toeplitz plus Hankel operators in 2003, the doors are open for the successful investigation of various classes of singular operators and related applications. This talk is devoted to constructive methods based upon explicit asymmetric factorization of matrix functions on the real line, which is quite different from the usual Wiener-Hopf factorization. A catalogue of possible consequences and new applications will be exposed and discussed.

The aim of the talk is to present a new approach to the investigation of the essential spectra of the main operators of quantum mechanics. We include these operators in a class of pseudodifferential operators perturbed by non-smooth potentials. For an operator under consideration we introduce a family of limit operators, and prove that the essential spectrum of the original operator is the union of spectra of limit operators. Since the limit operators have more simple structure than the original operator, we obtain a strong tool for the investigation of the essential spectra of differential and pseudodifferential operators. We apply this method to the study of the essential spectra of the Schrödinger, Klein-Gordon, and Dirac operators and to a new simple proof of the classical Hunziker-van Winter-Zjislin Theorem.