Contents/conteúdo

Mathematics Department Técnico Técnico

Functional Analysis and Applications Seminar  RSS

Sessions

Past

09/01/2015, 14:30 — 15:30 — Room P3.10, Mathematics Building
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.

Frank-Olme Speck 25/07/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
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.

Helena Mascarenhas 04/07/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
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 — Room P3.31, Mathematics Building
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 — Room P3.10, Mathematics Building
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.

Alexei Karlovich 04/04/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
, 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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 (u;v) of weights u and v for the boundedness of the maximal operator from a generalized local Morrey space with the weight u to another one with the weight v, 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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 a(x) on a bounded open set in the n-dimensional Euclidean space in the limiting Sobolev case a(x)p(x)=n is bounded from the variable exponent Lebesgue space to BMO, if p(x) satisfies the standard \(\log\)-condition and a(x) is Holder continuous of an arbitrarily small order.

05/07/2013, 14:30 — 15:30 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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 — Room P3.10, Mathematics Building
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.

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