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Patryk Pagacz

Publications and source records attributed to Patryk Pagacz.

14 recordsLinked to original sources
arXiv2025

On the dimension of orbits of matrix pencils under strict equivalence

We prove that, given two matrix pencils $L$ and $M$, if $M$ belongs to the closure of the orbit of $L$ under strict equivalence, then the dimension of the orbit of $M$ is smaller than or equal to the dimension of the orbit of $L$, and the equality is only attained when $M$ belongs to the orbit of $L$. Our proof uses only the majorization involving the eigenstructures of $L$ and $M$ which characterizes the inclusion relationship between orbit closures, together with the formula for the codimension of the orbit of a pencil in terms of its eigenstruture.

math.SP↗
arXiv2024

The Taylor spectrum of pairs of isometries

In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,\pi^2)$) of the closed bidisc.

math.SP↗
arXiv2024

An operator theory approach to the evanescent part of a two-parametric weak-stationary stochastic process

A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceed. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper we describe the most untouchable one - the evanescent part. Moreover, we point out how this part depends on the shape of the past.

math.PR↗
arXiv2024

On singular pencils with commuting coefficients

We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of its coefficients. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence we could propose a new description of (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.

math.SP↗
arXiv2024

On bundle closures of matrix pencils and matrix polynomials

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution of this paper is to prove that the dimension of each of these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The second main contribution is to prove that also the closure of the bundle of a matrix polynomial of grade larger than 1 is the union of the bundle itself with a finite number of other bundles of smaller dimension. To get these results we obtain a formula for the (co)dimension of the bundle of a matrix pencil in terms of the Weyr characteristics of the partial multiplicities of the eigenvalues and of the (left and right) minimal indices, and we provide a characterization for the inclusion relationship between the closures of two bundles of matrix polynomials of the same size and grade.

math.NA↗
arXiv2022

Matrix pencils with the numerical range equal to the whole complex plane

The main purpose of this article is to show that the numerical range of a linear pencil $λA + B$ is equal to $\mathbb{C}$ if and only if $0$ belongs to the convex hull of the joint numerical range of $A$ and $B$. We also prove that if the numerical range of a linear pencil $λA + B$ is equal to $\mathbb{C}$ and $A + A^*, B + B^* \geq 0$, then $A$ and $B$ have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.

math.NA↗
arXiv2021

Target Layer Regularization for Continual Learning Using Cramer-Wold Generator

We propose an effective regularization strategy (CW-TaLaR) for solving continual learning problems. It uses a penalizing term expressed by the Cramer-Wold distance between two probability distributions defined on a target layer of an underlying neural network that is shared by all tasks, and the simple architecture of the Cramer-Wold generator for modeling output data representation. Our strategy preserves target layer distribution while learning a new task but does not require remembering previous tasks' datasets. We perform experiments involving several common supervised frameworks, which prove the competitiveness of the CW-TaLaR method in comparison to a few existing state-of-the-art continual learning models.

cs.LG↗
arXiv2021

The Berberian's transform and an asymmetric Putnam-Fuglede theorem

We present how to apply a Berberian's technique to asymmetric Putnam-Fuglede theorems. In particular, we proved that if $A, B \in B(H)$ belong to the union of classes of $*$-paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and $AX = XB^*$ for some $X \in B(H)$, then $A^*X = XB$. Moreover, we gave a new counterexample for an asymmetric Putnam-Fuglede theorem for paranormal operators

math.FA↗
arXiv2020

Random Perturbations of Matrix Polynomials

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations.

math.PR↗
arXiv2020

Between the von Neumann inequality and the Crouzeix conjecture

A new concept of a deformed numerical range $W^ρ(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ ρ\in[1,+\infty)$ is a parameter. Each $W^ρ(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W^ρ(T)$ is decreasing with respect to $ ρ$ and $W^2(T)$ coincides with the numerical range. It is also shown that $W^ρ(T)$ is contained in the closed unit disc if and only if $T$ has a $ρ$ unitary dilation in the sense of Nágy-Foia\c s. The spectral constants of $W^ρ(T)$ are investigated, it is shown that it is monotone and continuous with respect to the parameter $ ρ$.

math.FA↗
arXiv2019

The decomposition theorems in Baer $*$-rings

We show a general decomposition theorem in Baer *-rings. As a consequence the vast majority of decompositions known in the algebra of bounded Hilbert space operators are generalized to Baer *-rings. There are also results which are new in the algebra of bounded Hilbert space operators. The model of summands in Wold-Słociński decomposition in Baer *-rings is given.

math.RA↗
arXiv2012

On the power-bounded operators of classes $C_{0 \cdot}$ and $C_{1 \cdot}$

By a bounded backward sequence of the operator $T$ we mean a bounded sequence $\{x_n\}$ satisfying $Tx_{n+1}=x_n$. In \cite{Pa} we have characterized contractions with strongly stable nonunitary part in terms of bounded backward sequences. The main purpose of this work is to extend that result to power-bounded operators. Aditionally, we show that a power-bounded operator is strongly stable ($C_{0 \cdot} $) if and only if its adjoint does not have any nonzero bounded backward sequence. Similarly, a power-bounded operator is non-vanishing ($C_{1 \cdot} $) if and only if its adjoint has a lot of bounded backward sequences.

math.FA↗