## About

44

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Introduction

Areas of interest: Toeplitz matrices and some classes of Toeplitz operators.
Personal page with presentations and interactive math visualizations:
http://www.egormaximenko.com.
Math lectures and exercises in simple Spanish:
http://esfm.egormaximenko.com.

Additional affiliations

Education

September 2000 - August 2003

## Publications

Publications (44)

The Jacobi-Trudi formulas imply that the minors of the banded Toeplitz matrices can be written as certain skew Schur polynomials. In 2012, Alexandersson expressed the corresponding skew partitions in terms of the indices of the struck-out rows and columns. In the present paper, we develop the same idea and obtain some new applications. First, we pr...

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as
the eigenvalues of a family of $n$-by-$n$ matrices as $n...

It is known that radial Toeplitz operators acting on a weighted Bergman space of the analytic functions on the unit ball generate a commutative C*-algebra. This algebra has been explicitly described via its identification with the C*-algebra \({{\rm VSO}(\mathbb{N})}\) of bounded very slowly oscillating sequences (these sequences was used by R. Sch...

We consider the set of all Toeplitz operators acting on the weighted Bergman space over the upper half-plane whose $L_∞$-symbols depend only on the argument of the polar coordinates. The main result states that the uniform closure of this set coincides with the C*-algebra generated by the above Toeplitz operators and is isometrically isomorphic to...

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram–Parter theorem. In the case of Hermitian matrices, the Avram–Parter theorem is equivalent to Szegő’s theorem on the eigenvalues. The Avram–Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the sing...

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re(α). After that, through the rest of the paper we su...

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $\alpha$ and the others of weight $1$. We denote by $n$ the order of the graph and suppose that $n$ tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on $\operatorname{Re}(\alpha)$. After that,...

Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumpt...

We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations \(\overline{D}^{(k_1,\ldots ,k_n)} f=0\) with \(k_1+\cdots +k_n=m\), can be written as polynomials of total degree \(<m\) in variables \(\overline{z_1},\ldots ,\overline{z_n}\), with some analytic coefficients. We establish a weighted mean va...

Let \(\mu _\alpha\) be the Lebesgue plane measure on the unit disk with the radial weight \(\frac{\alpha +1}{\pi }(1-|z|^2)^\alpha\). Denote by \({\mathcal {A}}^{2}_{n}\) the space of the n-analytic functions on the unit disk \({\mathbb {D}}\), square-integrable with respect to \(\mu _\alpha\). Extending results of Ramazanov (1999, 2002), we explai...

We introduce a concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibili...

In this paper we study the asymptotic behavior of the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, …, 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. This matrix family is a particular case of periodic Jacobi matrices. If |α|≤ 1, then the eigenvalues belong to [0, 4] and ar...

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that\[P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1})=Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}).\] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several familie...

We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations $\overline{D}^{(k_1,\ldots,k_n)} f=0$ with $k_1+\cdots+k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\ldots,\overline{z_n}$, with some analytic coefficients. We establish a weighted mean value prope...

Let $\mu_\alpha$ be the Lebesgue plane measure on the unit disk with the radial weight $\frac{\alpha+1}{\pi}(1-|z|^2)^\alpha$. Denote by $\mathcal{A}_n$ the space of the $n$-analytic functions on the unit disk $\mathbb{D}$, square integrable with respect to $\mu_\alpha$, and by $\mathcal{A}_{(n)}$ the true-$n$-analytic space defined as $\mathcal{A}...

In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-\alpha$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|\alpha|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,\pi]$...

This paper considers bounded linear radial operators on the polyanalytic Fock spaces ℱn and on the true-polyanalytic Fock spaces ℱ(n). The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in z and z¯. First, using this basis, we decompose the von...

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that \[ P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}) =Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}). \] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several fam...

The paper considers bounded linear radial operators on the polyanalytic Fock spaces $\mathcal{F}_n$ and on the true-polyanalytic Fock spaces $\mathcal{F}_{(n)}$. The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in $z$ and $\overline{z}$. Firs...

It was shown in a series of recent publications that the eigenvalues of n × n Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of n + 1. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol g(x) = (2 sin(x/2)...

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of th...

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin...

In a sequence of previous works with Albrecht Böttcher, we established higher-order uniform individual asymptotic formulas for the eigenvalues and eigenvectors of large Hermitian Toeplitz matrices generated by symbols satisfying the so-called simple-loop condition, which means that the symbol has only two intervals of monotonicity, its first deriva...

We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In this case, the Toeplitz localization operators (TLOs) exhibit an explicit diago...

In this paper we show that the C*-algebra generated by radial Toeplitz operators with L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty }$$\end{document}-symb...

The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. W...

We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In that case, the Toeplitz localization operators (TLOs) exhibit an explicit diago...

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results,...

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend only on the argument of the variable. This algebra is known to be commutative, and it is isometrically isomorphic to a certain algebra of bounded complex valued functions on the real numbers. In the paper we prove th...

En étendant le résultat récent de Herrera Yañez, Maximenko et Vasilevski, nous allons proposer une nouvelle étape dans lʼanalyse structurelle des algèbres générées par les opérateurs de Toeplitz agissant sur les espaces pondérés de Bergman sur le demi-plan supérieur. Nous allons montrer que lʼensemble des fonctions « spectrales » correspondant aux...

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend on the imaginary part of the argument only. Such algebra is known to be commutative, and is isometrically isomorphic to an algebra of bounded complex-valued functions on the positive half-line. In the paper we prove...

In the paper we deal with Toeplitz operators acting on the Bergman space A2(Bn) of square integrable analytic functions on the unit ball Bn in Cn. A bounded linear operator acting on the space A2(Bn) is called radial if it commutes with unitary changes of variables. Zhou, Chen, and Dong [9] showed that every radial operator S is diagonal with respe...

The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper c...

The paper is devoted to the asymptotic behavior of the eigenvectors of banded Hermitian Toeplitz matrices as the dimension
of the matrices increases to infinity. The main result, which is based on certain assumptions, describes the structure of
the eigenvectors in terms of the Laurent polynomial that generates the matrices up to an error term that...

In a recent paper, we established asymptotic formulas for the eigenvalues of the $n\times n$ truncations of certain infinite Hessenberg Toeplitz matrices as $n$ goes to infinity. The symbol of the Toeplitz matrices was of the form $a(t)=t^{-1}(1-t)^{\alpha}f(t)$ ($t\in{\mathbb T}$), where $\alpha$ is a positive real number but not an integer and $f...

a b s t r a c t While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are unifo...

While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known
about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n goes to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j...

The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications.
The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme
eigenvectors of the matrices as their dimension increases to infinity.

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense i...

The Szegő and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. We show that, surprisingly, these theorems are not true for every continuo...

We consider the limits of norms of inverse operators and pseudospectra convolution operators on expanding polyhedra.

We consider matrix convolution operators with integrable kernels on expanding polyhedra. We study their connection with convolution operators on the cones at the vertices of polyhedra. We prove that the norm of the inverse operator on a polyhedron tends to the maximum of the norms of the inverse operators on the cones, and the pseudospectrum tends...

## Projects

Project (1)

Consider an RKHS $H$ over a domain $X$ and a group action on $X$, such that the corresponding translation operators form a unitary representation $\rho$. Our goal is to describe the von Neumann algebra of all bounded linear operators commuting with $\rho$. The problem is solved for some particular cases using appropriate orthonormal bases, or appropriate Fourier transforms. For some particular cases, we describe the C*-algebra generated by the translation-invariant Toeplitz operators.