09/01/2015, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, Cuernavaca, México
On compactness of commutators of convolution type operators with PQC data
Applying the theory of Calderón- Zygmund operators, we study the compactness of commutators of multiplication operators $aI$ and convolution operators $W^0(b)$ on weighted Lebesgue spaces with Muckenhoupt weights on the real line for some clases of piecewise quasicontinuous functions $a,b$. Applications of these results are considered. The talk is based on a joint work with Isaac De la Cruz Rodríguez and Iván Loreto Hernández.
25/07/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, Universidade de Lisboa
Wiener-Hopf factorization through an intermediate space
An operator factorization conception is investigated for a general Wiener-Hopf operator $W = P_2 A|{P_1 X}$ in asymmetric Banach space setting. Namely, we study a particular factorization of the underlying operator $A = A_- C A_+$ where $A_+$ and $A_-$ are strong Wiener-Hopf factors and the cross factor $C$ maps an "intermediate space" $Z$ onto itself such that $Z$ is split into complemented subspaces closely related to the kernel and cokernel of $W$ and, moreover, such that $W$ is toplinear equivalent to a "simpler" symmetric Wiener-Hopf operator, $W \sim P C|_{PX}$.The main result shows equivalence between the generalized invertibility of the Wiener-Hopf operator and this kind of factorization (provided $P_1 \sim P_2$) which implies a formula for a generalized inverse of $W$. The conception embraces I.B. Simonenko's generalized factorization of matrix measurable functions in $L^p$ spaces and various other factorization approaches, particularly factorization of bounded into unbounded operators. It is quite different from the cross factorization approach and more useful in many applications. Some connected theoretical questions are answered such as: How to transform different kinds of factorization into each other? When is $W$ itself the truncation of a cross factor?
11/07/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, Cuernavaca, México
Index Calculation for Fredholm Singular Integral Operators with Shifts
The problem of index calculation for Fredholm singular integral operators with shifts and piecewise slowly oscillating data on Lebesgue spaces is considered. Corresponding Fredholm criteria are essentially based on the theory of Mellin pseudodifferential operators with non-regular symbols. The index study is based on joint works with V. G. Kravchenko, A. Yu. Karlovich and A. B. Lebre.
04/07/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Helena Mascarenhas, Instituto Superior Técnico
Variable-coefficient Toeplitz matrices and Singular Values
In this talk we describe asymptotic spectral properties of sequences of variable-coefficient Toeplitz matrices. These sequences, $A_N (a)$, with the symbol $a$ being in a Wiener type algebra and defined on a finite cylinder, widely generalizes the sequences of finite sections of a Toeplitz operator. We prove that if a does not vanish, then the singular values of $A_N (a)$ have the $k$-splliting property, which means that, there exist an integer $k$ such that, for $N$ large enough, the first $k$th-singular values of $A_N(a)$ converge to zero as $N$ goes to infinity while the others are bounded away from zero, with $k$ equals the sum of the kernel dimension of two Toeplitz operators.
The talk is based on joint work with B. Silbermann.
26/06/2014, 14:30 — 15:15 — Sala P3.10, Pavilhão de Matemática
Roland Duduchava, A. Razmadze Mathematical Institute
Calculus of tangential differential operators on hypersurfaces
Partial differential equations on surfaces in the Euclidean space and corresponding boundary value problems (BVPs), encounter rather often in applications. For example: heat conduction by a thin conductive surface or deformation a of thin elastic surface are governed by some differential equations on these surfaces. To rigorously derive equations which govern the above mentioned processes we need a calculus of tangential partial differential operators on a hypersurface (i.e., a surfaces in the Euclidean space $\mathbb{R}^n$ of co-dimension $1$). There are known many approaches to this problem, but the main task is to find the one which gives simplest results. We suggest a calculus of partial differential operators on a hypersurface based on Günter's and Stoke's tangential derivatives. We will expose basics of this calculus and show how classical differential operators, such as Laplace-Beltrami operator (governing the heat conduction), Lamé-Beltrami operator (governing the deformation of an elastic surface), Schrödinger equation, are written with the help of Günter's derivatives. We will end up with the demonstration of $\Gamma$-convergence of a BVP for the Laplace equation in a curved layer to a BVP for Laplace-Beltrami equation on a mid-surface when the thickness of the layer
20/06/2014, 14:30 — 15:30 — Sala P3.31, Pavilhão de Matemática
Ana C. Conceição, Universidade do Algarve
Analytical algorithms for computing the kernel of some classes of
singular integral operators
The main goal of this talk is to show how symbolic computation
can be used to compute the kernel of some classes of singular
integral operators. In addition, we present some results on the
dimension of the kernel of some classes of singular integral
operators whose kernel, in general, can not be determined in an
explicit form. The methods developed rely on innovative techniques
of Operator Theory and have a great potential of extension to more
complex and general problems. Some nontrivial examples computed
with the computer algebra system Mathematica are
presented.
06/06/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Pedro A. Santos, CEAF - Instituto Superior Técnico
Approximations of convolutions with almost periodic or
quasi-continuous symbol
We study the stability and Fredholm property of the finite
sections of convolution type
operators with semi-almost periodic and quasicontinuous
symbols, and operators of multiplication by slowly oscillating,
almost periodic or even more general coefficients. This is done by
introducing and exploring the notions of Rich sequences and
Quasi-Banded operators.
04/04/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade Nova de Lisboa
Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts
Let $\alpha$ and $\beta$ be orientation-preserving diffeomorphisms (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$, where the derivatives $\alpha'$ and $\beta'$ may have discontinuities of slowly oscillating type at $0$ and $\infty$. For $p\in(1,\infty)$, we consider the weighted shift operators $U_\alpha$ and $U_\beta$ given on the Lebesgue space $L^p(\mathbb{R}_+)$ by $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$ and $U_\beta f= (\beta')^{1/p}(f\circ\beta)$. We apply the theory of Mellin pseudodifferential operators with symbols of limited smoothness to study the simplest singular integral operators with two shifts $A_{ij}=U_\alpha^i P_++U_\beta^j P_-$ on the space $L^p(\mathbb{R}_+)$, where $P_\pm=(I\pm S)/2$ are operators associated to the Cauchy singular integral operator $S$, and $i,j\in\mathbb{Z}$. We prove that all $A_{ij}$ are Fredholm operators on $L^p(\mathbb{R}_+)$ and have zero indices. This is a joint work with Yuri Karlovich and Amarino Lebre.
10/01/2014, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, Cuernavaca, México
\(C^\ast\)-algebras of two-dimensional singular integral operators
with solvable groups of shifts
Fredholm symbol calculi for the \(C^\ast\)-algebras generated by
the \(C^\ast\)-algebra of two-dimensional singular integral
operators with continuous coefficients on a bounded closed simply
connected plane domain with Liapunov boundary and by unitary shift
operators associated with discrete solvable groups being semidirect
products of commutative groups of conformal mappings (elliptic,
hyperbolic or parabolic) and cyclic groups generated by reflections
are constructed. As a result, we establish Fredholm criteria for
the operators in considered algebras. The talk is based on a joint
work with V. Mozel.
20/12/2013, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Natasha Samko, Luleå University of Technology, Sweden and CEAF
Note on a two-weight estimate for the maximal operator in local
Morrey spaces
We obtain general type sufficient conditions and necessary
conditions on a pair of weights and for the
boundedness of the maximal operator from a generalized local Morrey
space with the weight to another one with the weight , with
some ”logarithmic gap” between these conditions. Both the
conditions formally coincide if we omit a certain logarithmic
factor in these conditions.
06/12/2013, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Eugene Shargorodsky, King's College London, UK
On the level sets of the resolvent norm of a linear operator
In 1976, J. Globevnik posed a question on whether or not the
resolvent norm of a bounded linear operator on a Banach space can
be constant on an open set. The question remained open until 2008.
The talk is a survey of recent results and open questions related
to Globevnik's question.
25/10/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve and CEAF
A BMO-result for potential operators in the variable exponent
We show that the Riesz fractional integration operator of variable
order on a bounded open set in the -dimensional Euclidean
space in the limiting Sobolev case is bounded from
the variable exponent Lebesgue space to BMO, if satisfies
the standard \(\log\)-condition and is Holder continuous of
an arbitrarily small order.
05/07/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, Cuernavaca, México
A boundary value problem with a finite group of Lipschitz shifts
Fredholm criteria and index formulas for singular integral
operators with piecewise slowly oscillating coefficients and
finite non-cyclic groups of Lipschitz shifts whose derivatives
admit slowly oscillating discontinuities are established by
applying the theory of Mellin pseudodifferential operators with
non-regular symbols. Such operators studied on the Lebesgue spaces
are related to boundary value problems with finite groups of
shifts.
21/06/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Nuno António, Instituto Superior Técnico and CEAF
Trigonometric \(\operatorname{sl}(2)\) Gaudin model with boundary
terms
Starting from the non-symmetric \(R\)-matrix of the inhomogeneous
\(XXZ\) spin-\(1/2 \) chain and generic solutions of the reflection
equation and the dual reflection equation, the corresponding Gaudin
Hamiltonians with boundary terms are derived. An alternative
derivation based on the so-called classical reflection equation is
discussed.
10/05/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Stefan Samko, Universidade do Algarve and CEAF
Morrey spaces and Stummel classes
We prove a new property of Morrey function spaces: local Morrey
type behaviour of functions is very close to weighted behaviour.
More precisely, generalized local Morrey spaces are embedded
between weighted Lebesgue spaces with weights differing only by a
logarithmic factor. This leads to the statement that the
generalized global Morrey spaces are embedded between two
generalized Stummel classes whose characteristics similarly differ
by a logarithmic factor. We give examples proving that these
embeddings are strict. For the generalized Stummel spaces we also
give an equivalent norm.
12/04/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Frank Speck, Instituto Superior Técnico, UTL and CEAF
On the reduction of linear systems related to boundary value
problems
The main topic of this work is the investigation of operator
relations which appear during the reduction of linear systems,
particularly in the study of boundary value problems. The first
objective is to improve formulations like "equivalent reduction" by
the help of operator relations. Then we describe how some of these
operator relations can be employed to determine the regularity
class and effective solution of boundary value problems.
Furthermore operator relations are used to put boundary value
problems into a correct space setting, e.g., by operator
normalization.
15/03/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Luís Pessoa, Instituto Superior Técnico, UTL and CEAF
The Essential Boundary on Polyanalytic Functions. Some Differences
Between the Analytic and the Polyanalytic Cases.
I will begin to explain how the action of the compression of the
Beurling transform on the Bergman space can give a transparent view
of the structure of poly-Bergman spaces on domains Möbius
Equivalent to a Disk. The existence of exact Dhzuraev formulas is
an important tool. Note that the usual representations of
poly-Bergman projections by means of two-dimensional singular
integral operators are strongly dependent on the smoothness of the
boundary. In the second part of the talk, I will consider a bounded
domain without constrains on the boundary. A Fredholm symbolic
calculus is constructed for poly-Toeplitz operators with continuous
symbol and I will explain how such symbol requires the notion of
j-essential boundary. The symbol calculus is well known for
Bergman-Toeplitz operators, in which setting the removal boundary
is a subset of the boundary having zero logarithmic capacity. Some
surprising differences between the analytical and the
poly-analytical case will be presented.
18/01/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Torsten Ehrhardt, University of California, Santa Cruz, USA
Resultant matrices and inversion of Bezoutians
The subject of this talk are special types of structured matrices.
The inversion of finite Toeplitz matrices is very well studied, and
the inverses of Toeplitz matrices are so-called Bezout matrices. We
pursue to opposite goal, the inversion of (invertible) Bezout
matrices. Special attention is paid to explicit formulas and a fast
computation of the inverse. It turns out that the problem is
related to another problem, namely the description of the kernel of
generalized resultant matrices. This problem is studied as well.
The talk is based on joint work with Karla Rost.
11/01/2013, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, Cuernavaca, México
\(C^*\)-algebras of singular integral operators with shifts
admitting distinct fixed points
Fredholm symbol calculi for the \(C^*\)-algebras
\(\mathfrak{B}\) of singular integral operators with piecewise
slowly oscillating coefficients extended by groups of unitary shift
operators are constructed. The groups of unitary shift operators
are associated with discrete amenable groups of piecewise smooth
homeomorphisms that act topologically freely on the unit circle and
admit distinct fixed points for different shifts. As a result,
faithful representations of the quotient \(C^*\)-algebras
\(\mathfrak{B}/{\mathfrak{K}}\), where \({\mathfrak {K}}\) is the
ideal of compact operators, on suitable Hilbert spaces are
constructed by applying the local-trajectory method, spectral
measures and a lifting theorem, and Fredholm criteria for the
operators \(B\in\mathfrak{B}\) are established.
The talk is based on a joint work with M. A. Bastos and C. A.
Fernandes.
14/12/2012, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Alexandre Almeida, Universidade de Aveiro
Riesz and Wolff potentials in variable exponent weak Lebesgue
spaces and applications
We study mapping properties of variable order Riesz and Wolff
potentials in variable exponent weak Lebesgue spaces. It is of
special interest the case when these potentials act on \(L^1\). We
also discuss how the results can be applied to the study of
integrability properties of solutions of some elliptic equations.