04/10/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T.L.
Norms of Toeplitz and Hankel Operators on Hardy Type Subspacesof
Rearrangement-Invariant Spaces
We prove analogues of the Brown-Halmos and Nehari theorems on the
norms of Toeplitz and Hankel operators, respectively, acting on
subspaces of Hardy type of reflexive rearrangement-invariant spaces
with nontrivial Boyd indices over the unit circle.
13/09/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Bernd Silbermann, Technische Universität Chemnitz
Collocation Matrices and Computation of Partial Indices of Regular
Matrix Functions
The talk is aimed at the computation of the partial indices of
regular matrix functions knowing the values of these functions at
distinguished points. The proposed algorithm is not only stable but
also converges sufficiently fast provided the underlying function
is smooth.
26/07/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yuri I. Karlovich, Universidad Autónoma del Estado de Morelos, México
An Index Formula for Toeplitz Operators with Oscillating Symbols
The talk is devoted to calculating the indices of Toeplitz
operators with oscillating matrix symbols on the Hardy space over
the upper half-plane. The symbols belong to the -algebra
generated by semi-almost periodic and slowly oscillating matrix
functions. An approach which does not use the harmonic extensions
of slowly oscillating matrix functions is presented.
19/07/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Semyon Yakubovich, Universidade do Porto
Recent Results on Some Integral Operators Associated with theKontorovich-Lebedev Transformation
Further developments of the results on the Kontorovich-Lebedev type transformations and associated convolution in $L_p$ spaces are given. In particular, we consider compositions with the Mellin type convolution transformations and obtain some interesting cases of integral transforms depending upon parameters of hypergeometric functions (the Lommel functions, the Whittaker functions, the Clausenian functions, etc.). Boundedness properties are investigated, Plancherel type theorems are proved. Parseval type identities are established. Applications to integral equations of the Fredholm and convolution type associated with the Kontorovich-Lebedev operator are given.
12/07/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Luís Pessoa, Instituto Superior Técnico, U.T.L.
Álgebra Gerada pelas Projecções de Bergman e Anti-Bergman comCoeficientes Seccionalmente Contínuos
Usando localização estabelecem-se critérios de Fredholm para a álgebra-$C^\ast$ gerada pelas projecções de Bergman e anti-Bergman com coeficientes seccionalmente contínuos no semi-plano superior. As secções consideradas são definidas por curvas regulares tais que em pontos da fronteira do semi-plano coincidem numa vizinhança com segmentos de recta. A hipótese anterior relaciona-se com a aplicação dum trabalho de Plamenevsky, baseado numa decomposição da transformada de Fourier multidimensional. A estratégia cria a necessidade de estabelecer uma particular caracterização de certas álgebras-$C^\ast$ geradas por projecções ortogonais em espaços de Hilbert. É igualmente estabelecido, usando $\ast$-isomorfismos locais, um $\ast$-isomorfismo entre a álgebra considerada e uma correspondente no disco unitário. O seminário é baseado num trabalho conjunto com Y. Karlovich.
05/07/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Nenad Manojlovic, Universidade do Algarve, Faro
Trigonometric $\operatorname{osp}(1|2)$ Gaudin Model
The problems connected with Gaudin models are reviewed by analyzing models related to the trigonometric $\operatorname{osp}(1|2)$ classical $r$-matrix. Moreover, the eigenvectors of the trigonometric $\operatorname{osp}(1|2)$ Gaudin Hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik-Zamolodchikov system is discussed both in the rational and the trigonometric case.
07/06/2002, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Juan Carlos Sanchez Rodriguez, Universidade do Algarve, Faro
Sobre a Factorização de Algumas Classes de Funções Matriciais
24/05/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Maria Teresa Alzugaray, Universidade do Algarve, Faro
Singularidades de Funções Geradoras de Sucessões Frequenciais dePólya
Uma sucessão chama-se sucessão frequencial de Pólya (ou multiplamente positiva) de ordem $r$, se todos os menores de ordem menor ou igual a $r$ (todos os menores se $r$ é igual a $\infty$) da sua matriz de Toeplitz são não negativos. A nossa comunicação estará relacionada com as singularidades de funcões geradoras dessas sucessões (f.g.'s PFr) para $r$ finito. Para cada $r$ finito, descreveremos, em termos de ordens próximas, o possível crescimento que uma f.g. PFr pode ter no seu círculo de convergência e perto das suas singularidades. Serão ainda descritos todos os possíveis domínios de holomorfia de f.g.'s PFr. Para a obtenção dos resultados acima mencionados foram desenvolvidos dois novos métodos de construção de f.g.'s PFr não inteiras.
17/05/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T.L.
Transformation Techniques towards the Factorization of Non-Rational
2 x 2 Matrix Functions
For the Wiener-Hopf factorization of matrix
functions defined on a closed Carleson curve, so-called rational
transformations, i.e. multiplication by rational matrix functions,
are important. In the first part of the lecture, the topic will be
motivated by a variety of applications and by general operator
theoretical facts as well. Then we establish a classification
scheme for matrix functions, which is based on such
transformations. We determine invariants under these
transformations and describe those matrix functions which can be
transformed to triangular or Daniele-Khrapkov form. In the third
part we consider special rational transformations and study the
same problem. For instance, we consider transformations with matrix
functions that are analytic and invertible on an open neighborhood
of the given curve.
The talk is based upon common work with Torsten Ehrhardt, to be
published in Linear Algebra and Its Applications under the
same title.
03/05/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Maria Amélia Bastos, Instituto Superior Técnico, U.T.L.
Fredholm Theory in an Algebra Generated by OperatorswithOscillating
Symbols and the Cauchy Singular Operator
In this talk we deal with a Fredholm theory in the algebra
generated by operators with oscillating symbols and the Cauchy
singular operator as a subalgebra of bounded linear operators in
(-vector) Lebesgue spaces.
In the first part of the talk, using the limit operator theory,
a necessary Fredholmness condition for any operator in is
established and, from one of the local principles, sufficient
Fredholmness conditions for some elements in this algebra are
obtained. A Fredholm criterion for Toeplitz operators with matrix
symbols in the algebra generated by slowly oscillating
and semi-almost periodic matrix functions on is
established as a consequence.
In the second part, using the notion of harmonic extension, an
index theory for Fredholm Toeplitz operators whose generating
functions belong to the algebra is developed. The most
relevant result is obtained by reducing to Toeplitz operators whose
generating functions belong to the space of semi-almost periodic
matrix functions.
This talk is based on common work with Y. Karlovich and B.
Silbermann.
19/04/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Instituto Superior Técnico, U.T.L.
Invertibility in Banach Algebras of Functional Operators
withNon-Carleman Shifts
We prove the inverse closedness of the Banach algebra of
functional operators with non-Carleman shifts, which have only two
fixed points, in the Banach algebra of all bounded linear operators
on . We suppose that the generators of the algebra have
essentially bounded data. An invertibility criterion for functional
operators in is obtained in terms of the invertibility of a
family of discrete operators on . An effective invertibility
criterion is established for binomial difference operators with
bounded coefficients on the spaces . Using the reduction to
binomial difference operators, we give effective criteria of
invertibility for binomial functional operators on the spaces
.
These results are obtained in collaboration with Yuri
Karlovich.
05/04/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Martin Edwards, The Queen's College, Oxford, England, UK
Algebraic Structure of Complex Banach Spaces
A complex Banach space possesses an intrinsic algebraic structure
associated with the holomorphic properties of its open unit ball.
This introduction will give describe the algebraic structure in a
variety of examples, and will also give examples to show how the
holomorphic, geometric and algebraic properties of complex Banach
spaces are interwoven.
22/03/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve, Faro
The Singular Type Operators in the Lebesgue Spaces with Variable Exponent
Last decade there was intensively developed the theory of Lebesgue spaces with variable exponent when the order $p$ of integrability depends on $x$. (These spaces have interesting applications in fluid mechanics). The corresponding theory proved to be difficult to develop because these spaces are not invariant neither with respect to translation nor dilation. It suffices to mention that for example Young type theorems for convolutions are not already valid in these spaces. In general, convolution operators have a "bad" behaviour in such spaces. A progress was recently made by proving the uniform boundedness of dilation convolution operators under some natural assumptions on the kernel of the convolution. This result, presented in particular, in the talk allowed us to prove that "nice" functions (infinitely differentiable with compact support) are dense not only in the Lebesgue spaces with variable exponent, but also in Sobolev spaces generated by them. However, boundedness of the singular integral operators remained an open question for a long time. We show that some modification of the method developed in the above investigation allows us to prove also that the singular operator along a bounded Lyapunov curve is bounded in the space with a variable exponent $p(x)$ under some natural assumptions on $p(x)$. The last topic of the talk is based on the joint research with Prof. Vakhtang Kokilashvili.
12/03/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Israel Gohberg, Tel Aviv University, Israel
Infinite Systems of Linear Equations
Infinite systems of linear equations are usually solved by the
method of finite sections. This means that the infinite system is
replaced by the sequence of finite sections of the original system
and it is expected that the solutions of the finite systems
converge to the solution of the infinite system. This method has a
rich history of 150 years and many distinguished mathematicians
have made important contributions in this area. The talk will
present the early history and recent important achievements.
Unexpected examples and computational experiments will motivate and
illustrate the main results. Special attention will be paid to the
case of Toeplitz matrices with continuous and discontinuous
symbols.
08/03/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Vakhtang Kokilashvili, Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia
The Boundary Value Problems for Analytic and Harmonic Functions
The Dirichlet and Neumann problems for harmonic functions from the
Smirnov type classes in domains with arbitrary piecewise smooth
boundaries will be discussed. The picture of solvability is
described completely; the non-Fredholm cases are exposed; an
influence of geometric properties of boundaries on the solvability
is revealed; a criterion for the unique solvability of this problem
is established for arbitrary boundary values from the Lebesque
spaces with exponent greater than one. Similar problems are
considered in weighted Smirnov classes of harmonic functions. The
weight is an arbitrary power function. In the classes of harmonic
functions which are real parts of analytic functions represented in
the domains by the Cauchy type integrals we investigated the
Dirichlet problem with boundary functions from the weighted Zygmund
classes. In all the cases of solvability there are given explicit
formulas for the solution in terms of Cauchy type integrals and
conformal mapping functions. The proofs are heavily based on the
investigation of the linear conjugation problem with oscillating
conjugation coefficient, the boundary properties of derivatives of
functions which map conformally the unit circle onto a domain with
an arbitrary piecewise smooth boundary and two-weighted norm
inequalities for singular integrals. The talk is based on joint
papers by V.Kokilashvili, V. Paatashvili and Z.Meshveliani.
07/02/2002, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
Perturbed Toeplitz Operators with Oscillating Symbols
25/01/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Sergei V. Rogosin, Belarussian State University, Minsk
Newton-Kantorovich Method
for Conformal Representation of Plane Domains
11/01/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
António Bravo, Instituto Superior Técnico, U.T.L.
Operadores Integrais Singulares com Deslocamento Carlemaniano e
Coeficientes Oscilantes
14/12/2001, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Levan Sigua, I. Javakhishvili Tbilisi State University, Georgia
Screen magnetic type problems for the vector
Helmholtz equation