05/06/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Leiba Rodman, College of William and Mary, Williamsburg, VA, USA
Linear dependence of operators via sesquilinear forms
The numerical values (associated with the numerical ranges) and
\(q\)-numerical values (associated with the \(q\)-numerical ranges)
of two Hilbert space operators are compared. The main result of the
talk states that the absolute values of the \(q\)-numerical value
function of two operators coincide if and only if the operators are
unimodular scalar multiples of each other, for \(q\) positive and
less than one. In the extreme cases when \(q\) is equal to one or
to zero, additional possibilities occur. These statements are
framed in terms of \(C\)-numerical ranges where the operator \(C\)
is nonscalar of rank one. The results are motivated by an
application to the problem (still largely unsolved) of
characterizing norm preservers of Jordan products of matrices.
(Joint work with B. Kuzma, G. Lesnjak, C.-K. Li, T. Petek.)
24/04/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade Nova de Lisboa and CEAF, IST, UT Lisboa
Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves
In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(G)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^p(G)$ where $p(\cdot)$ satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.
17/04/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ilya Spitkovsky, College of William and Mary, Williamsburg, VA, USA
Matrices with normal defect one
A \(n\times n\) matrix \(A\) has normal defect one if it is not
normal, however can be embedded as a north-western block into a
normal matrix of size \((n+1)\times (n+1)\). The latter is called a
minimal normal completion of \(A\). A construction of all matrices
with normal defect one is given. Also, a simple procedure is
presented which allows one to check whether a given matrix has
normal defect one, and if this is the case, to construct all its
minimal normal completions. A characterization of the generic case
for each n under the assumption that the rank of the
self-commutator of \(A\) equals \(2\) (which is necessary for \(A\)
to have normal defect one) is obtained. Both the complex and the
real cases are considered. It is pointed out how these results can
be used to solve the minimal commuting completion problem in the
classes of pairs of \(n \times n\) Hermitian (resp., symmetric, or
symmetric/antisymmetric) matrices when the completed matrices are
sought of size \((n+1)\times (n+1)\). An application to the
\(2\times n\) separability problem in quantum computing is
described. This is a joint work with Dmitry Kaliuzhnyi-Verbovetskyi
and Hugo Wourdeman.
03/04/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ilya Spitkovsky, College of William and Mary, Williamsburg, VA, USA
On common invariant cones for families of matrices
The existence and construction of common invariant cones for
families of real matrices is considered. The complete results are
obtained for \(2\times 2\) matrices (with no additional
restrictions) and for families of simultaneously diagonalizable
matrices of any size. Families of matrices with a shared dominant
eigenvector are considered under some additional conditions. This
is a joint work with Leiba Rodman and Hakan Seyalioglu.
13/02/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Chafiq Benhida, Université de Lille 1, France
Hyponormality and subscalarity
Starting from M. Putinar's result showing that a hyponormal operator has a scalar extension which means that it is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoara-Foias), we discuss this notion and show that backward Aluthge iterates of hyponormal operators are subscalar.
06/02/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Natasha G. Samko, Centro de Análise Funcional e Aplicações, Faro
Weighted boundedness of singular operators in Morrey spaces
We study the problem of weighted boundedness of the one-dimensional singular operator with Cauchy kernel in Morrey spaces on a curve. The weight function may be a product of a finite number of almost monotonic functions with nodes on the curve. The boundedness of the operator in case of such a weight is reduced to the boundedness of Hardy type operators in Morrey spaces with this weight. We prove the latter, which enables us to obtain sufficient conditions for the boundedness of the singular operator in terms of the Matuszewska-Orlicz indices of weights. A special attention is paid to the case of power weights where we prove that the conditions on the weight are also necessary for the boundedness. We also discuss an application to the study of Fredholmness of singular integral equations in weighted Morrey spaces, the interest to this investigation being caused by the non-separability of Morrey spaces.
05/12/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade Nova de Lisboa and CEAF, IST, UT Lisboa
Connectedness of spectra of Toeplitz operators on Hardy spaces with Muckenhoupt weights over Carleson curves
Harold Widom proved in 1966 that the spectrum of a Toeplitz operator $T(a)$ acting on the Hardy space $H^p(T)$ over the unit circle $T$ is a connected subset of the complex plane for every bounded measurable symbol $a$ and $p \gt 1$. In 1972, Ronald Douglas established the connectedness of the essential spectrum of $T(a)$ on $H^2(T)$. We show that, as was suspected, these results remain valid in the setting of Hardy spaces $H^p(G,w), p \gt 1,$ with general Muckenhoupt weights $w$ over arbitrary Carleson curves $G$. This is a joint work with Ilya Spitkovsky.
14/11/2008, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Luís Castro, Universidade de Aveiro, Portugal
Solvability of singular integro-differential equations with multiple complex shifts
We will consider initial value problems for functional equations on the half-axis that contain Hilbert transforms, derivatives and complex shifts. The class of problems is motivated by various applications, and will be considered in both Bessel potential and Sobolev–Slobodeckij space settings. Results are Fredholm and invertibility criteria as well as explicit analytical solution in cases where techniques for the constructive factorization of symbol matrices are available. The talk is based on a joint work with R. Duduchava and F.-O. Speck.
14/11/2008, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Ioannis Stratis, National and Kapodistrian University of Athens, Greece
Transmission and boundary value problems in the theory of elasticity of hemitropic materials
A material is called hemitropic (acentric, noncentrosymmetric, chiral) if it is isotropic with respect to orthogonal transformations, but not with respect to mirror reflections. Typical examples are quartz crystals, DNA, bones and nanotubes. Although engineers have studied hemitropic materials since the early 1960s, the theory of elasticity for hemitropic materials has only very recently become the object of rigorous mathematical analysis. The governing equations describing time harmonic elastic fields in hemitropic materials, constitute a system of two vector elliptic PDEs of second order, stated in terms of the displacement and the microrotation vectors. When all the “hemitropicity” parameters become zero, this system reduces to the well-known reduced Navier equation of classical (isotropic) linear elasticity. In this talk, we present the basic boundary value and transmission problems arising in 3-dimensional linear hemitropic elasticity. We construct the fundamental solution that satisfies a Sommerfeld-Kupradze type radiation condition. We establish Green’s formulae, based on which we obtain representations of solutions (in bounded and unbounded domains) in terms of appropriate potentials (single-layer, double-layer, Newtonian). We study the jump and mapping properties of these potentials, and of the corresponding boundary integral (pseudodifferential) operators. Using Potential Theory and the Theory of Pseudodifferential Operators we establish the uniqueness and the existence of solutions (in Hölder, Sobolev and Besov spaces) to the Dirichlet, Neumann, and mixed type boundary value problems and to transmission problems. Further, we study the regularity properties of the solutions to these problems. The solvability is treated in both the cases of smooth and of Lipschitz domains.
07/11/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Franciszek Szafraniec, Universytet Jagiellonski, Krakow, Poland
Unbounded subnormal operators: the highlights
I intend to provide as much as the time permits with the basic facts of theory of unbounded subnormal operators. This is a class of operators living somehow aside, out of the main interest of operator theorist, presumably because of lack standard tools. My goal is to convince the audience that this is much undeserved. I hope some distinguised example from quantum mechanics will help me to fulfil the promise.
31/10/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Catarina Carvalho, Instituto Superior Técnico, U.T. Lisboa
Topological index for pseudodifferential operators on $\mathbb{R}^n$
We develop a pseudodifferential calculus on $\mathbb{R}^n$ suitable to the study of the Fredholm index from the topological viewpoint, namely to generalizations of the index formula on compact spaces. We consider operators that are multiplication by a matrix valued function outside a compact set and give some properties of this class. We then study the closure of a suitable subalgebra. In particular, we find explicit Fredholm criteria and show that the symbols of Fredholm operators have similar topogical properties to those on compact spaces, leading in the same way to a topological formula for the index. Our results have applications on wider classes of pseudodifferential operators on $\mathbb{R}^n$ , namely the isotropic and scattering calculi, and also to pseudodifferential operators on non-compact manifolds.
17/10/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Mihaly Bakonyi, Centro de Estruturas Lineares e Combinatorias, Universidade de Lisboa
Factorizations of operator-valued functions on ordered groups
Sz. Nagy and Foias used an approach based upon the Wold decomposition of an isometry for proving factorization results for operator-valued functions on the unit circle. We are applying an analogue of the Wold decomposition for semigroups of isometries to give some geometric insight into factorization results by Helson and Lowdenslager for matrix-valued functions defined on compact groups with a totally ordered dual. By using some counterexamples, we show that the extensions of these results to operator- valued functions face some basic obstructions. The talk is based on joint work with Dan Timotin (Mathematical Institute of the Romanian Academy).
26/09/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade Nova de Lisboa
Maximal operators on variable Lebesgue spaces with weights related
to oscillations of Carleson curves
We prove sufficient conditions for the boundedness of the
maximal operator on variable Lebesgue spaces with weights
, where is a complex number, over
arbitrary Carleson curves. If the curve has different spirality
indices at the point and is not real, then the weight
is an oscillating weight lying beyond the class of radial
oscillating weights considered recently by V. Kokilashvili, N.
Samko, and S. Samko.
19/09/2008, 15:15 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ivan Todorov, QueenŽs University Belfast, United Kingdom
\(s\)-numbers of elementary operators
A well-known theorem of Fong and Sourour states that an
elementary operator acting on the space of all bounded
linear operators on a Hilbert space is compact if and only if the
symbols of the operator can be chosen to be compact. In this talk
we will give quantitative versions of this result using the notions
of an \(s\)-number function introduced by A. Pietsch and the theory
of ideals of developed by von Neumann, R. Schatten and W.
Calkin. We will relate the behaviour of the -numbers of a given
elementary operator to that of its symbols. We will further extend
these results to the case of elementary operators acting on general
C*-algebras. The talk is based on a joint work with M. Anoussis and
V. Felouzis.
19/09/2008, 14:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Bernd Silbermann, Technische Universität Chemnitz, Germany
100 years Galerkin's method
There are some reasons to assume that Galerkin's method (or what
is called now Galerkin's method) was born about 100 years ago. It
is not quite clear to whom one has to adress the priority, but
without doubt, Bubnov, Galerkin, Ritz and Simic belong to the
circle of main actors. Interestingly enough, the first idea of
Galerkin's method was created in order to solve approximately some
biharmonic problems occuring in the theory of thin plates. I shall
try to describe in short a part of these developments which then
had a considerable influence on forming Numerical Mathematics both
for biharmonic problems and yet for more general settings.
Subsequently I will mention some theoretical concepts of projection
and more general approximation methods for solving operator
equations and then pass back to biharmonic problems. Especially I
will discuss a method of approximate solution of such problems
based on function theory considerations.
11/07/2008, 15:15 — 16:00 — Sala P3.10, Pavilhão de Matemática
Matthew Heath, Instituto Superior Técnico, U.T. Lisboa
Compact failure of multiplicativity for linear maps between Banach
algebras
The definition of compactness (and that of weak compactness) for
a linear map between normed spaces may be extended to multilinear
maps in a fairly natural way. We treat compactness as a sort of
"smallness" condition for multilinear maps. For Banach algebras
and we call a linear map, , a
cf-homomorphism (meaning "compact from a homomorphism") if the
bilinear map , (i.e. if the "failure to be multiplicative") is a compact
bilinear map. We give general theorems showing that such maps are
rather well behaved as well as numerous examples. In particular we
characterise the pairs of compact, Hausdorff spaces and for
which cf-isomorphisms from to are automatically
multiplicative.
11/07/2008, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Yuri I. Karlovich, Universidad Autónoma del Estado de Morelos, Mexico
Wiener-Hopf operators with symbols generated by semi-almost periodic and slowly oscillating matrix functions
The talk is devoted to studying the Wiener-Hopf operators with symbols generated by semi-almost periodic and slowly oscillating matrix functions with entries in the Banach algebra of Fourier multipliers on weighted Lebesgue spaces. Fredholm and invertibility results are obtained. The talk is based on a joint work with Juan Loreto Hernández.
20/06/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Lina Oliveira, Instituto Superior Técnico, U.T. Lisboa
Range tripotents and order
The partial order relation existing in the set of tripotents in a *-triple is such that it suffices to adjoin a greatest element for the new set to become a complete lattice. For example, such is the case of the partial isometries in a *-algebra which, as it is well-known, are exactly the tripotents in the algebra. Any *-subtriple of determines a subset of tripotents called the range tripotents relative to . It is the aim of this talk to present new results concerning the restricted partial order relation in . Furthermore, an analysis of the suprema of families of range tripotents will lead to establishing a counterpart of a result already existing for projections in *-algebras. It is intended to provide the background material needed to make the talk understandable to an audience familiar with operator algebras.
06/06/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Giorgi Bogveradze, Andrea Razmadze Mathematical Institute, Tbilisi, Georgia
Invertibility characterization of scalar Wiener-Hopf plus Hankel
operators with essentially bounded Fourier symbols
The invertibility of Wiener-Hopf plus Hankel operators with
essentially bounded Fourier symbols is characterized via certain
factorization properties of the Fourier symbols. In addition, a
Fredholm criterion for these operators is also obtained and the
dimensions of the kernel and cokernel are described. The talk is
based on a joint paper with L. P. Castro.
30/05/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
El Hassan Zerouali, Faculté des Sciences de Rabat, Maroc
Hypercyclic transform of operators
Let be a bounded operator on some Hilbert space and be its Aluthge transform. and share in general their spectral properties and have been intensively treated this last decade. In our talk the cyclic behaviour is investigated. We show that adjoints have he same cyclic behaviour. This is done through some commutation techniques.