# Functional Analysis, Linear Structures and Applications Seminar

## Past sessions

### $C^\ast$-algebras of Bergman type operators with piecewise slowly oscillating coefficients

Given a simply connected domain $U$ in the complex plane with a piecewise Dini-smooth boundary which admits a finite set of Dini-smooth corners, we study the $C^\ast$-algebra $B_U$ generated by the Bergman and anti-Bergman projections acting on the Lebesgue space $L^2(U)$ and by the operators of multiplication by piecewise continuous functions that slowly oscillate at points of the domain boundary. Applying the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on $\operatorname{VMO}_{\partial}$ functions, we construct a Fredholm symbol calculus for the $C^\ast$-algebra $B_U$ and establish a Fredholm criterion for the operators $A\in B_U$.

The talk is based on joint work with E. Espinoza-Loyola.

### Fourier-type operators, new convolutions and their applicability to integral equations

Considering new generalizations of the Fourier transform, besides studying their general properties, we construct new convolutions associated with such operators. Moreover, for the convolutions we obtain corresponding factorization identities and some norm inequalities. These convolutions allow us to consider new classes of integral equations and to study the solvability of different types of integral equations. Conditions for the existence of unique solutions of those equations are presented, as well as the corresponding explicit solutions.

### Symplectic eigenvalues of positive definite matrices

For every positive definite matrix of order $2n$, there exist $n$ positive numbers associated to it. These numbers are called the symplectic eigenvalues of the matrix, and are important in different areas such as classical (Hamiltonian) mechanics, quantum information and symplectic topology. Recently there has been a heightened interest in the study of symplectic eigenvalues both by mathematicians and physicists due to their extensive applications in quantum information. In this talk, we discuss some fundamental inequalities and variational principles involving symplectic eigenvalues, perturbation theorems and relationship between symplectic eigenvalues and ordinary eigenvalues.

### Spectrum of the Robin-Laplacian: recent results, and open problems

I will discuss two recent theorems and one recent conjecture about maximizing or minimizing the first three eigenvalues of the Robin Laplacian of a simply connected planar domain. Conformal mappings and winding numbers play a key role in the geometric constructions. In physical terms, these eigenvalues represent decay rates for heat flow assuming a “partially insulating” boundary.

This is joint with Geometry Seminar.

### On a class of Integral operators in central generalized Morrey spaces

We find conditions for the boundedness of integral operators $K$ which commute with dilations and rotations, in a central generalized Morrey space. We also show that under the same conditions these operators preserve the subspace of Morrey spaces, known as vanishing Morrey space. In the case of non-negative kernels, we also give necessary conditions for the boundedness. In the case of classical Morrey spaces the obtained sufficient and necessary conditions coincide with each other. In the one-dimensional case we also obtain similar results for global Morrey spaces. In the case of radial kernels we obtain stronger estimates of $Kf$ via spherical means of $f$. We demonstrate the efficiency of the obtained conditions for a variety of examples such as weighted Hardy operators, weighted Hilbert operator, their multi-dimensional versions and others.

### Combinatorial Game Theory: a survey

In the first years of the 20th century the analysis of the game of NIM (by the mathematician Charles L. Bouton) triggered the outburst of a completely new mathematical subject: Combinatorial Game Theory. The aim of this seminar is to give a survey of the development of this mathematical field.

### Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by H. Mascarenhas, P. Santos and M. Seidel. The preprint is available at https://arxiv.org/abs/1903.09696.

### The asymptotic behaviour of the Super-Plancherel measure

Let $\mathbb{F}_q$ be a finite field and denote $U_n(\mathbb{F}_q)$ the group of $n \times n$ uppertriangular matrices over $\mathbb{F}_q$ with only ones in the diagonal. In recent years the representation theory of $U_n(\mathbb{F}_q)$ has been approached via certain Supercharacter-Theories, not only due to their (non-commutative) combinatorial-analogues to the representation theory of the symmetric group $S_n$, but also as a useful tool to address Harmonic-analysis problems.

We consider a particular Supercharacter-Theory for $U_n(\mathbb{F}_q)$ which yields a natural measure on the set-partitions of $\{1,...,n \}$: the Super-plancherel measure $\textbf{SPl}_n$. The aim of this talk is to understand the asymptotic behaviour of $\textbf{SPl}_n$ as $n \rightarrow \infty$; in particular limit objects are interpreted in a representation-theoretical setting.

### Minors of Toeplitz matrices, symmetric functions and random matrix ensembles

We study minors of Toeplitz matrices, using the theory of symmetric functions. The symbols of the Toeplitz matrices can either be in the Szegö class or have Fisher-Hartwig singularities and the results apply both to minors of finite and of large dimension. As an application, also using results on inverses of Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials. We also discuss the intimate relationship with random matrix theory and the ensuing applications.

### A geometric model for the module category of a gentle algebra

One of the problems in representation theory is to understand the structure of the module category of an algebra. Gentle algebras are a class of algebras of tame representation type, meaning it is often possible to get a global understanding of their representation theory.

In this talk, we will show how to encode the module category of a gentle algebra using combinatorics of a surface. This is joint work with Karin Baur (Graz).

### Involutions for symmetries of Littlewood-Richardson (LR) coefficients

The action of the dihedral group $\mathbb Z_2\times \mathfrak{ S}_3$ on LR coefficients is considered with the action of $\mathbb Z_2$ realized by the transposition of a partition. The action of $\mathbb Z_2\times\mathfrak{ S}_3$ carries a linear time action of a subgroup $H$ of index two, and a bijection which goes from $H$ into the other coset is difficult. LR coefficients are preserved in linear time by the action of $H$ whereas the other half symmetries consisting of commutativity and transposition symmetries are hidden. The latter are given by a remaining generator, the reversal involution or Schutzenberger involution, by which one is able to reduce in linear time the LR commuters and LR transposers to each other.

To pass from symmetries of LR (skew) tableaux to symmetries of companion Gelfand-Tsetlin (GT) patterns we build on the crystal action of the longest permutation in the symmetric group on an LR tableau, and Lascoux's double crystal graph structure on biwords. This analysis also affords an explicit bijection between two of the interlocking GT patterns in a hive.

This is based on a joint work with A. Conflitti and R. Mamede.

### Toeplitz matrices and Schur polynomials

Toeplitz matrices are ubiquitous and enjoy attractive computational properties, for which its study has captured the interest of many mathematicians in the last century.

In the first part of this talk, we review some necessary properties of Schur polynomials. We start a well known tool, Vieta’s formulas, from which naturally arise the elementary symmetric polynomials, then we study other families of symmetric polynomials: complete homogeneous symmetric polynomials, Schur polynomials, and skew Schur polynomials. We also recall a formula that expresses the product of Schur polynomials in terms of skew Schur polynomials.

In the second part, we give relations between Schur polynomials and Toeplitz banded matrices, for instance, we express determinants and minors of Toeplitz matrices through Schur polynomials.

The results presented here have been obtained jointly with Egor Maximenko; they are based on some ideas by William F. Trench and Per Alexandersson.

The speaker has been partially supported by IPN-SIP projects.

### The spectral determinant of the quantum harmonic oscillator in arbitrary dimensions

We show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity. We determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large.

As a consequence, we provide an alternative proof of a conjecture posed by Bar and Schopka concerning the convergence of the determinant of the Dirac operator on $S^{n}$, determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the proof by Moller.

The seminar is joint with Geometria em Lisboa seminar.

### Four concrete applications of Toeplitz operators

I present four concrete and very different applications of Toeplitz operators.

These applications are (1) a problem in optimal ell-one control, (2) Wiener-Hopf and spectral factorization of polynomials of degree 20000, (3) computation of the volume of the fundamental domains of some high-dimensional lattices, and (4) the determination of the Hausdorff limit of the zero set of polynomials of the Fibonacci type. The talk allows you to switch off four times and to re-enter the same number of times.

### Robinson-Schensted and RSK correspondences for Skew and Skew Shifted Tableaux

The Robinson-Schensted correspondence, introduced by Schensted (1961) in its most well-known form, presents a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. Knuth (1970) presented a generalization, the RSK correspondence, for semistandard Young tableaux.

Asthe Young tableaux are deeply tied to the study of linear representations of the symmetric group, there is a variant for the projective representations, the shifted tableaux, based on strict partitions. The shifted tableaux also arise in the study of $Q$-functions, an important basis for the subalgebra of the symmetric functions generated by the odd power sums, which resembles the classic Schur functions.

In this seminar, we will show generalizations of both Robinson-Schensted and RSK correspondences for skew tableaux and analogues for shifted skew tableaux (Sagan, Stanley, 1990), based on variants of the insertion algorithm. We will also present a brief introduction on some aspects of the theory of shifted tableaux.

### One-sided invertibility of discrete functional operators with bounded coefficients

The one sided invertibility of discrete functional operators with bounded coefficients on the spaces $l^p(\mathbb{Z})$ with $p\in[1,\infty]$ is studied. Criteria of the one sided invertibility of such operators generalize those obtained in the case of slowly oscillating behavior of coefficients. Criteria of the one-sided invertibility of discrete functional operators associated with infinite slant-dominated matrices are established. Applications to studying the two- and one-sided invertibility of functional operators on Lebesgue spaces are also considered.

### On the spectrum of the Hilbert matrix operator

For each, $\lambda\in\mathbb{C}$, $\lambda\neq 0,-1,-2,...$ the (generalized) Hilbert matrix $\mathcal{H}_{\lambda}$ is given by $$\mathcal{H}_{\lambda}:=\left(\frac{1}{n+m+\lambda}\right)_{n,m\geq0}.$$ If $\lambda=1$ then $\mathcal{H}_{\lambda}$ is the classical Hilbert matrix introduced by D. Hilbert about 125 years ago. These matrices have been the subject of numerous investigations. The talk mainly concerns the description of spectral properties of Hankel operators generated by these matrices on the Hardy spaces $H^{p}$ and $l^{p}$ $(1 < p < \infty$). Special attention will be paid to the description of the essential and point spectra of these operators.

### On a simple proof of the boundedness of Bergman projection in various Banach spaces and some related questions

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of V. P. Zaharyuta, V. I. Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey space. In the case of variable exponent Lebesgue space the boundedness results is known, so in that case we provide a simpler proof. The other two cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis.

We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

This is a joint work with Stefan Samko and Humberto Rafeiro.

### Integral Graphs

A graph $G$ is called integral if all eigenvalues of its adjacency matrix, $A(G)$, consist entirely of integers. The nullity of $G$ is the nullity of $A(G)$, that is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this talk, we are concerned with integral trees. These objects are extremely rare and very difficult to find. We first present a short survey on integral graphs. We show that for any integer $d \gt 1$, there are infinitely many integral trees of diameter $d$. We will also show that for any integer $k \gt 1$, there are only finitely many integral trees with nullity $k$.

### Algebras of convolution type operators with PSO data

The Fredholm symbol calculus is constructed for the Banach algebras generated on the weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights by all multiplication operators by piecewise slowly oscillating functions and by all convolution operators with piecewise slowly oscillating presymbols being Fourier multipliers. A new approach of identifying local spectra is presented, which allows one to complete the description of the Fredholm symbol calculus.

Older session pages: Previous 2 3 4 Oldest

Current organizers: Helena Mascarenhas, Ângela Mestre.