Seminario de Análisis y Aplicaciones UAM-ICMAT 2019

20 de Marzo de 2020, 10:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Glenier Bello, UAM-ICMAT
Models of linear operators satisfying operator inequalities.PDF,

A general spectral theory of a linear operator does not exist, it only exists for
particular subclasses of operators. One of the most celebrated theories of this
type is the Nagy-Foiaş spectral theory of Hilbert space contractions, obtained
in the 1960ies. It is based on the construction of a functional model, which
heavily relies on certain chapters of Complex Analysis, such as the theory of
Hardy spaces.
Being a contraction on a Hilbert space is characterized by a very simple
operator inequality. In a landmark work of 1982, Agler showed how to pass
from some other operator inequalities to a functional model of an operator
by applying more general reproducing kernel Hilbert spaces instead of Hardy
spaces. This work motivated an extensive research, which has become a rapidly
growing branch of Operator Theory.
I will explain how to construct an explicit functional model for an operator
satisfying a rather general operator inequality, and discuss the uniqueness of
this model. Some spectral consequences in the spirit of the Nagy-Foiaş theory
will be derived. I will also explain a new connection with the ergodic theory of
linear operators.

13 de Marzo de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Pablo Berná, Universidad CEU San Pablo
¿Podemos ser avariciosos en espacios quasi-Banach?PDF,

Desde el año 1999, la teoría de aproximación avariciosa (o greedy) se ha
desarrollado en el contexto de espacios de Banach, aunque algunos autores
han dado por hecho en el mundo quasi-Banach ciertos resultados que se
tenían en el caso Banach. La intención de esta charla es analizar si es trivial
o no la extensión del mundo de bases greedy en espacios de Banach al caso
quasi-Banach. Además, daremos algunos resultados que estamos desarrollando
sobre espacios de aproximaciÃ³n con pesos generales en combinación
con las llamadas bases semi-greedy. Los resultados que mostraremos forman
parte de dos trabajos conjuntos con F. Albiac, J. L. Ansorena, E. Hernández y
P. Wojtaszczyk.

6 de Marzo de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
María Jesús Carro, Universidad Complutense de Madrid
Solving problems in ergodic theory via restricted weak type extrapolation.PDF,

The main goal of this talk is to present several applications in the setting
of ergodic theory related with the Return Time Theorem of Bourgain via of an
extension of Rubio de Francia extrapolation theorem in the setting of weighted
restricted weak type inequalities.

28 de Febrero de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Ivana Slamić, University of Rijeka
Maximal cyclic subspaces for dual integrable representations.PDF,

Consider a countable discrete group acting on a separable Hilbert space
via unitary represenation. If such representation is dual integrable, then the
structure of the invariant subspaces and various properties of orbits can be
analyzed using the corresponding bracket map. As a special case, we obtain
systems of integer translates of a square integrable function. The properties of
these systems have been extensively studied and among the known results is
the fact that such system is \(\ell^2\) -linearly independent precisely when the
periodization function is positive a.e. On the other hand, this condition is
equivalent to maximality of the principal shift-invariant subspace which the system
generates. Characterization of other levels of linear independence is, in most cases,
still an open problem in general, however, we know that the equivalence with
maximality no longer holds if we replace \(\ell^2\) with \(\ell^p\)-linear independence, for
\(p\neq 2\). In this talk, after briefly recalling the main results and questions
concerning this topic, we shall focus on several questions related with maximal cyclic
subspaces for the previously described group setting, which are the part of a
recent research in collaboration with H. Šikić.

21 de Febrero de 2020, 11:30 : Aula Naranja, ICMAT
Pedro Tradacete, ICMAT
Strictly singular operators between \(L^p\) spaces.PDF,

Recall that an operator between Banach spaces is strictly singular provided
it is not invertible when restricted to any (closed) infinite dimensional subspace.
The class of strictly singular operators forms a closed two-sided operator ideal,
containing compact operators, and was introduced by T. Kato in connection
with the perturbation theory of Fredholm operators. In this talk we will focus on
the interpolation properties of this class of operators acting between different
\(L^p\) spaces, and the structure of strictly singular non-compact operators. In
particular, by means of Riesz potential operators acting between measure spaces
of different Hausdorff dimension, we will see that the set of pairs \((1/p,1/q)\)
such that an operator is strictly singular but not compact from \(L^p\) to \(L^q\)
can contain a line segment of any positive slope.
The talk is based on joint work with F. L. Hernández and E. M. Semenov.

14 de Febrero de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Diana Carbajal, Universidad de Buenos Aires
Diagonalization of Shift-Preserving Operators.PDF,

In this talk we discuss the structure of bounded shift-preserving operatorsacting on shift-invariant spaces of
\(L^2(\mathbb{R}^d)\). For this, we work with an isometry called fiberization map and study the properties that
the correspondent range function and range operator induce. We introduce a new notion of diagonalization for
these operators which we call s-diagonalization and give a generalized Spectral Theorem for normal
shift-preserving operators. Finally, we apply these results to a dynamical sampling problem.
This work is in collaboration with A. Aguilera, C. Cabrelli and V. Paternostro.

31 de Enero de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Fernando Gómez Cubillo, Universidad de Valladolid
On covariant frames and coherent states.PDF,

Covariant frames and coherent states associated with unitary
representations of locally compact groups are of interest in signal theory and
mathematical physics. The talk introduces these concepts in harmonic analysis and their
relationship with convolution Hilbert algebras and weights on the group von
Neumann algebras.

24 de Enero de 2020, 11:30 : Módulo 17, Aula 520, Depto. Matemáticas UAM
Alex Amenta, Universität Bonn
Vector-valued time-frequency analysis and the bilinear Hilbert transform.PDF,

The bilinear Hilbert transform is a bilinear singular integral operator (or
Fourier multiplier) which is invariant not only under translations and dilations, but
also under modulations. This additional symmetry turns out to make proving \(L^p\) -
bounds especially difficult. I will give an overview of how time-frequency
analysis is used in proving these \(L^p\)-bounds, with focus on the recently-understood
setting of functions valued in UMD Banach spaces.

17 de Enero de 2020, 11:30 : Aula Naranja, ICMAT
Oscar Domínguez, Universidad Complutense de Madrid
New estimates for maximal functions.PDF,

Maximal functions play a central role in the study of differentiation, singular
integrals and almost everywhere convergence. With Sergey Tikhonov (ICREA)
we recently proved some pointwise estimates for maximal functions in terms
of smoothness and rearrangements. I plan to discuss the recent progress on
these topics and some applications. In particular, I will discuss the
Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which
are close to \(L^\infty\) .