arXiv subjects

Oskar Jakub Szymański

Publications and source records attributed to Oskar Jakub Szymański.

3 recordsLinked to original sources

arXiv2025

The Szász inequality for matrix polynomials and functional calculus

The Szász inequality is a classical result that provides a bound for polynomials with zeros in the upper half of the complex plane, expressed in terms of their low-order coefficients. Generalizations of this result to polynomials in several variables have been obtained by Borcea-Brändén and Knese. In this article, we discuss the Szász inequality in the context of polynomials with matrix coefficients or matrix variables. In the latter case, the estimation provided by the Szász-type inequality can be sharper than that offered by the von Neumann inequality. As a byproduct, we improve the scalar Szász inequality by relaxing the assumption regarding the location of zeros.

math.FA↗

arXiv2024

Stability Theory for Matrix Polynomials in One and Several Variables with Extensions of Classical Theorems

The file contains PhD Dissertation by Oskar Jakub Szymański. This work ends his study at Doctoral School of Exact and Natural Sciences at Jagiellonian University where the Author has attended in years 2019-2024. The subject of the Thesis is the stability theory developed for matrix polynomials. The main concept described in the Dissertation is hyperstability with respect to an open or closed subset of the complex plane. The notion of hyperstability is a contribution of the Author of the Thesis. Disseration is devoted mostly to regular polynomials, singular polynomials play marginal role in the Thesis. The Author of the Thesis has discovered several interesting results extending classical theorems from complex analysis. These are for example: multi-variable Gauss-Lucas theorem for matrix polynomials and Szász-type inequalities for matrix norms such as the matrix two-norm and the Frobenius norm.

math.CV↗

arXiv2022

Stability Of Matrix Polynomials In One And Several Variables

The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several second- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.

math.CV↗