29/06/2007, 15:15 — 16:15 — Room P3.10, Mathematics Building
Ana Paula Nolasco, Universidade de Aveiro
A Fredholm characterization for Wiener-Hopf-Hankel operators
withsemi-almost periodic symbols
It will be presented a characterization of Fredholm and
invertibility properties for Wiener-Hopf-Hankel operators with
semi-almost periodic Fourier symbols and acting between
Lebesgue spaces, based on the mean motions and geometric mean
values of the almost periodic representatives of the Fourier
symbols at minus and plus infinity. Additionally, a formula for the
Fredholm index is derived, as well as some conditions for the
invertibility of the operators.
29/06/2007, 14:00 — 15:00 — Room P3.10, Mathematics Building
Ilya Spitkovsky, College of William and Mary, Williamsburg, Virginia, USA
Spectra of some Toeplitz operators with almost periodic matrix
symbols
We will discuss the current state of the factorization problem for
almost periodic matrix functions and the consequences it has for
the spectral theory of related Toeplitz operators. Some open
problems will be presented as well.
01/06/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Centro de Análise Funcional e Aplicações, Lisboa
Semi-Fredholm singular integral operators with piecewise continuous
coefficients on weighted variable Lebesgue spaces are Fredholm
Kokilashvili, Paatashvili, and Samko proved in 2005 that the Cauchy
singular integral operator is bounded on variable Lebesgue spaces
with Khvedelidze weights on arbitrary Carleson curves. We show that
if the Carleson curve satisfies, in addition, a so-called
logarithmic whirl condition at each point, then every semi-Fredholm
operator in the Banach algebra of singular integral operators with
matrix piecesise continuous coefficients is Fredholm.
18/05/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Amarino Lebre, Instituto Superior Técnico, U.T. Lisboa
Factorization of singular integral operators with a
Carlemanbackward shift: the case of bounded measurable coefficients
This talk is based on a joint work with V. G. Kravchenko and J.
S. Rodríguez. We consider a generalization of the results of a
recent work by the authors concerning scalar singular integral
operators with a backward Carleman shift, allowing more general
coefficients, bounded measurable functions on the unit circle. The
main purpose is to obtain, for singular integral operators with a
backward shift and bounded measurable coefficients, an operator
factorization from which the Fredholm characteristics, like the
kernel and the cokernel, can be described. The main tool is the
factorization of matrix functions. In the course of the analysis
performed for that class of operators several useful
representations are obtained which permit, in particular, to
completely characterize the set of invertible operators in that
class, providing explicit examples of such operators.
04/05/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Juan Carlos Sanchez Rodríguez, Universidade do Algarve, Faro
Factorization of singular integral operators with a Carleman
backward shift: the case of continuous coefficients
It is well known that when dealing with (pure) singular integral
operators on the unit circle with coefficients belonging to a
decomposing algebra of continuous functions, a factorization of the
symbol induces a factorization of the original operator, which is a
representation of the operator as a product of three singular
integral operators where the outer operators in that representation
are invertible. In our seminar we will show a similar operator
factorization for the case of singular integral operators with a
backward shift. We also show that the factorization of the
considered operators is related to a (special) factorization in a
algebra of block diagonal matrix functions and that such operator
factorization is also possible for other classes of singular
integral operators, namely those including either a conjugation
operator or a composition of a conjugation with a forward shift
operator.
27/04/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Natasha G. Samko, Centro de Análise Funcional e Aplicações, Faro
Indices of almost monotonic functions depending on a parameter and their applications to Hölder spaces of variable order
Hölder spaces of variable order $\lambda(x)$ varying from point to point are well known. Meanwhile, it is also possible to consider generalized Hölder spaces with a characteristic $\omega(h)=\omega(x,h)$ which may also depend on the point $x$, similarly to the case of a Musielak-Orlicz space with the Young function $\Omega(x,u)$ depending on the point $x$. Since the Zygmund-Bary-Stechkin classes of characteristics $\omega$ are described in terms of the so called index numbers of $\omega$, in this generalization we arrive at indices depending on the parameter $x$ (this parameter in general may belong to an arbitrary abstract set, in applications this set may be a set in metric measure spaces). In this talk we consider properties of such parameter dependent index numbers, one of the main points being the study of conditions under which the Zygmund type inequality for $\omega(x,\cdot)$ is uniform with respect to the parameter. We shall discuss an application to measuring local dimensions of a metric measure space at a point $x$ and give an application to the study of generalized Hölder spaces on the unit sphere in the Euclidean space.
20/04/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Stefan G. Samko, Universidade do Algarve, Faro
On variable exponent analysis on homogeneous spaces
We start from a short survey of known results in Harmonic Analysis in Lebesgue spaces on metric measure spaces with doubling condition (homogeneous spaces) in the case of constant , this case having a long history, and in the case of variable , for this case only a single result on non-weighted boundedness of the maximal operator being recently obtained. After that we give a new result on the weighted boundedness of maximal Hardy-Littlewood operator on homogeneous metric measure spaces, for variable . We prove this result under certain sufficient condition which we call an "ersatz" of the Muckenhoupt condition. This sufficient condition nevertheless coincides with the necessary Muckenhoupt condition when is constant. A special class of weights is also considered, which includes "radial type" weights oscillating between two power functions (Zygmund-Bary-Stechkin type functions). In this case a stronger statement on the weighted boundeness is obtained. In connection with the weighted boundedness, we also introduce a new notion of lower and upper local dimensions of metric measure spaces. The talk is based on the joint work with Prof. Vakhtang Kokilashvili.
16/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa
Asymptotics of Toeplitz matrices with symbols in generalized Hölder
spaces
We study asymptotics of block Toeplitz matrices generated by matrix
symbols with entires in a generalized Hölder space or in the
closure of smooth functions in the generalized Hölder space. We
specify the speed of convergence in the Szegö-Widom limit theorems
and refine corresponding results of Böttcher and Silbermann for
standard Hölder spaces. Wiener-Hopf factorization in decomposing
algebras plays a crucial role in our investigation.
09/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Anatoli Merzon, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico
About the scattering of plane waves by wedges
We consider a nonstationary scattering of plane waves by a
wedge. It is assumed that the incident wave does not depend on the
coordinate parallel to the the edge of the wedge, so the problem is
planar. Also we assume that, beginning with a certain time instant
depending on a spatial position of the point, the incident wave is
periodic in time with the frequency \(\omega\) in each point of the
space. Let the profile of the wave be such that the incident wave
has the front ahead of which it is zero. Therefore the incident
wave establishes a harmonic vibration at any point of the
complement of the wedge with the frequency \(\omega\). The main
goal is to prove that the amplitude of the solution to the
corresponding mixed problem for the D'Alembert equation with
initial data determined by the incident wave, tends to the
solutions of the classical stationary diffraction problem. Thus,
these classical solutions can be represented as the limiting
amplitudes of the solutions to the non-stationary problem, i.e. the
Limiting Amplitude Principle holds. It is proved for the Dirichlet
and Neumann boundary conditions and for Dirichlet-Neumann boundary
conditions only for the right angle.
02/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roland Duduchava, A. Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia
Boundary value problems for shell equations
We propose writing partial differential equations on a hypersurface
in cartesian coordinates of the ambient space instead of more
customary local coordinates and the Riemannian metric tensor of the
underlying surface. This seemingly trivial idea simplifies the form
of many classical differential equations on the surface
(Laplace-Beltrami, Lamé, Maxwell etc.), which turn out to have
constant coefficients, and enables more transparent proofs of
Korn's inequalities, tightly connected with solvability and
uniqueness of some boundary value problems. The obtained results
are applied to the Dirichlet and Neumann boundary value problems
for the Laplace-Beltrami operator, for its square, and for the
elasticity Lamé operators, describing thin shells in the form of an
open smooth hypersurface with smooth boundary. An explicit Green
formula is derived and it is proved that the Dirichlet boundary
value problems has a unique solution in the Sobolev space of weak
solutions while the Neumann boundary value problems are solvable
under the usual orthogonality constraints on the data.
26/01/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Teresa Malheiro, Universidade do Minho, Guimarães
Meromorphic factorization revisited and application to a group of matrices
It is shown that the symbols in a class of exponentials of nilpotent matrices can be reduced, by splitting some rational factors, to a very simple normal form. A meromorphic factorization [1] of these symbols is thus naturally defined and by approaching the problem of transforming it into a generalized factorization from a point of view different from that of [1], we study the invertibility and the Fredholm properties of the Toeplitz operators with symbols in that class. These results simplify and generalize those obtained in [2]. This is a joint work with Cristina Câmara, IST, Lisboa. - Câmara, M. C., Lebre, A., Speck, F.-O. Generalised factorisation for a class of Jones form matrix functions. Proc. Roy. Soc. Ed., 123A (1993) 401-422.
- Câmara, M. C., Malheiro, M. T. Wiener-Hopf factorization for a group of exponentials of nilpotent matrices. Linear Algebra Appl. 320(1-3) (2000) 79-96.
19/01/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Catarina Carvalho, Instituto Superior Técnico, U.T. Lisboa
The Fredholm index in -theory for -algebras
In this talk, I will give an introduction to index theory in the non-commutative geometry framework. -theory for -algebras will be developed, and it will be shown how one can associate -theory classes to Fredholm operators, so that the Fredholm index arises as a map in -theory. We will see also how this approach leads to generalized Fredholm indices, which do not take integer values, but instead take values in some -theory group. One example of particular importance is that of families of operators.
12/01/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa
Generalized Krein algebras and asymptotics of Toeplitz determinants
We give a survey on generalized Krein algebras and Wiener-Hopf
factorization in these algebras. We discuss the role of generalized
Krein algebras in the asymptotic theory of Toeplitz determinants.
We pay special attention to some cases of "insuffucient
smoothness", when the Szegö-Widom limit theorem requires a higher
order correction involving additional terms and regularized
operator determinants. This talk is based on a joint work with
Albrecht Böttcher and Bernd Silbermann.
15/12/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
Algebras of singular integral operators with shifts
The talk is devoted to studying pseudodifferential operators with
non-regular symbols and their applications to singular integral
operators with shifts. Algebras of singular integral operators with
discrete subexponential groups of shifts are studied on weighted
Lebesgue spaces provided that the contour, the weight, the
coefficients and the shifts are slowly oscillating. Applying a
local-trajectory method and a theory of Mellin pseudodifferential
operators with non-regular symbols, we construct Fredholm symbol
calculi for the mentioned algebras of singular integral operators
with shifts and establish corresponding Fredholm criteria.
14/12/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Ernst Stephan, Universität Hannover, Alemanha
Corner singularities and Mellin symbols
Firstly, we show mapping properties (within countably normed spaces) of several boundary integral operators acting on polygons (e.g. the weakly singular single layer potential operator for the Laplacian and the corresponding double layer potential operator). Our analysis of the boundary integral equations is based on Mellin techniques and uses the Mellin symbols of the integral operators. Functions (with corner singularities but) belonging to countably normed spaces can be approximated very efficiently by the -version of the boundary element method when refining the mesh size and increasing the polynomial degree . Secondly, we analyze the Dirichlet problem for the Laplacian in a polygonal domain. Applying Mellin techniques to the boundary integral equation we show that the solution has a decomposition into regular and singular parts which blow up at certain exceptional angles. We derive a modified decomposition which depends continuously on the angle and can be used efficiently for boundary element computations.
30/11/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Vakhtang Kokilashvili, A. Razmadze Mathematical Institute, Tbilisi, Georgia
Singular integrals in weighted Lebesgue spaces with variable
exponent and applications
The goal of the talk is to present the boundedness criteria for
singular integrals in weighted Lebesgue spaces with variable
exponent. The following topics will be discussed: boundedness of
the generalized singular integrals on Carleson curves arising in
the theory of I. N. Vekua's generalized analytic functions;
sufficient conditions and examples of a couple of weights ensuring
two-weight estimates for singular integrals and maximal functions
in variable Lebesgue spaces; applications to the Dirichlet boundary
value problem in "bad" domains and to the problem of the mean
summability of Fourier trigonometric series in non-standard,
two-weight setting.
24/11/2006, 15:15 — 16:15 — Room P3.10, Mathematics Building
Alexey N. Karapetyants, CINVESTAV, Mexico
Toeplitz operators with special symbols in weighted Bergman spaces
We study Toeplitz operators in a weighted Bergman space on the unit
disc with a power type weight related to the boundary of the disc.
We deal with special symbols connected to the three types of
hyperbolic geometry in the unit disc (elliptic, parabolic and
hyperbolic pencils). That is, in each of the mentioned three cases
the symbols are constant on geodesics orthogonal to the
trajectories forming a pencil. The spectrum of each of the Toeplitz
operator seems to be quite accidental, the definite tendency starts
appearing only as the exponent of the weight tends to infinity. The
correspondence principle (F. Berezin) suggests that the limit set
of those spectra has to be strictly connected with the range of the
initial symbol. This is definitely true for continuous symbols.
Given a continuous symbol a, the limit set of spectra does coincide
with the range of a. The new effects appear when we consider more
complicated symbols. In particular, in the case of piecewise
continuous symbols the limit set coincides with the range of a
together with the line segments connecting the one-sided limit
points of piecewise continuous symbol. Note that these additional
line segments may essentially enlarge the limit set comparing to
the range of a symbol.
24/11/2006, 14:00 — 15:00 — Room P3.10, Mathematics Building
António Ferreira dos Santos, Instituto Superior Técnico, U.T. Lisboa
Lax equations, factorization and Riemann-Hilbert problems
In this talk the problem of existence and calculation of solutions
to Lax equations that define finite-dimensional integrable systems
is studied. The method presented is based on Wiener-Hopf
factorization and related Riemann-Hilbert problems on Riemann
surfaces. The idea behind the method was first proposed by
Semenov-Tian-Shansky but has never been applied in a situation
where a nontrivial function factorization is required. An example
of a dynamical system associated with an elliptic curve is given.
10/11/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
On the analytical solution of the linear-fractional Riemann problem
The linear-fractional problem is a generalization of the linear
Riemann problem that includes the (non-linear) factorization
problem. In case of normal type it can be equivalently reduced to a
family of homogeneous linear vector Riemann problems by space
foliation and adequate substitutions. Moreover these problems are
equivalent to systems of non-homogeneous Toeplitz equations with
special data. The reduced problem can be solved by matrix
factorization in various cases. This research is based upon joint
work with S. V. Rogosin.
03/11/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Sofia Naique, Instituto Superior Técnico, U.T. Lisboa
Polynomial almost periodic solutions for a class of
Riemann-Hilbertproblems
We consider a class of Riemann-Hilbert problems with triangular
symbols. Our investigation is devoted to the existence and
calculation of a solution, in the form of an almost periodic
polynomial. The Fourier spectrum of a solution of this kind is a
subset of a particular additive group. A necessary and sufficient
condition for the existence of a solution is obtained. Indeed, it
is a simple condition on the Fourier spectrum of the matrix symbol.
Explicit solutions are also determined, for different classes of
Riemann-Hilbert problems, which are determined once again by the
matrix symbol.