Fast non-Hermitian Toeplitz eigenvalue computations, joining matrix-less algorithms and FDE approximation matrices

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and we know that the eigenvalue computation is a non-trivial task in the non-Hermitian setting for large sizes. We follow the work of Bogoya, B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently, and combined the results to produce a high precision global numerical and matrix-less algorithm. The numerical results are very precise and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when all the spectrum is computed. From the viewpoint of real world applications, we emphasize that the matrix class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final conclusion section a concise discussion on the matter and few open problems are presented.Comment: 21 page

Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case

Abstract. We consider Toeplitz operators T (λ) a acting on the weighted Bergman spaces A2 λ (Π), λ ∈ [0, ∞), over the upper half-plane Π, whose symbols depend on θ = arg z. Motivated by the Berezin quantization procedure we study the dependence of the properties of such operators on the parameter of the weight λ and, in particular, under the limit λ → ∞. 1

Toeplitz operators with radial symbols on weighted holomorphic Orlicz space

Peer reviewe

Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $\alpha\in\mathbb{C}$, with $\operatorname{Re}(\alpha)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(\alpha) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(\alpha)<0$ and $n>\operatorname{Re}(\alpha-1)/\operatorname{Re}(\alpha)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(\alpha)^2/(2\operatorname{Re}(\alpha)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms.Comment: 28 pages, 8 figure

Generalized Electromagnetic fields in Chiral Medium

The time dependent Dirac-Maxwell's Equations in presence of electric and magnetic sources are written in chiral media and the solutions for the classical problem are obtained in unique simple and consistent manner. The quaternion reformulation of generalized electromagnetic fields in chiral media has also been developed in compact, simple and consistent manner

Commutative C∗C^*-algebras of Toeplitz operators on complex projective spaces

We prove the existence of commutative $C^*$-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space $\mathbb{P}^n(\mathbb{C})$. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of $\mathbb{P}^n(\mathbb{C})$

Toeplitz operators on the domain {Z∈M2×2(C)∣ZZ∗<I}\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\} with U(2)×T2\mathrm{U}(2)\times\mathbb{T}^2-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators

Quaternion Analysis for Generalized Electromagnetic Fields of Dyons in Isotropic Medium

Quaternion analysis of time dependent Maxwell's equations in presence of electric and magnetic charges has been developed and the solutions for the classical problem of moving charges (electric and magnetic) are obtained in unique, simple and consistent manner

Generalized Gravi-Electromagnetism

A self consistant and manifestly covariant theory for the dynamics of four charges (masses) (namely electric, magnetic, gravitational, Heavisidian) has been developed in simple, compact and consistent manner. Starting with an invariant Lagrangian density and its quaternionic representation, we have obtained the consistent field equation for the dynamics of four charges. It has been shown that the present reformulation reproduces the dynamics of individual charges (masses) in the absence of other charge (masses) as well as the generalized theory of dyons (gravito - dyons) in the absence gravito - dyons (dyons). key words: dyons, gravito - dyons, quaternion PACS NO: 14.80H