Applications of the Sylvester operator in the space of slice semi-regular functions
type: Item
instance of: scholarly article
title: Applications of the Sylvester operator in the space of slice semi-regular functions
zbMATH Open document ID: 1450.30068
author: Amedeo Altavilla
author: Chiara de Fabritiis
published in: Concrete Operators
publication date: 2020-04-23T00:00:00Z
review text: Let \(\Omega\) be an axially symmetric domain contained in the skew algebra \(\mathbf{H}\) of quaternions. Let \(\mathcal{R}(\Omega)\) denote the family of slice regular functions on \(\Omega\). A \(\ast\)-product is introduced for functions in \(\mathcal{R}(\Omega)\) whereby along with pointwise sums for addition \(\mathcal{R}(\Omega)\) becomes an associative algebra. Using these operations the authors study the behavior of functions in \(\mathcal{R}\mathcal{M}(\Omega)\) the set of slice semi-regular functions. Heavily relying on results of the authors in [``Equivalence of slice semi-regular functions via Sylvester operators'', Preprint, \url{arXiv:1907.07385}], three main results emerge along with several corollaries. First the authors show for \( f,g\) in \(\mathcal{R}\mathcal{M}(\Omega)\backslash \{0\}\) the kernel of the linear operator \(\mathcal{L}_{f,g}(\mathcal{X})=f^*\mathcal{X}^\ast{g}\) has dimension 2 if and only if exactly one between \( f\) and \( g\) is a zero divisor, while it has dimension 3 if and only if \(f\) and \(g\) are both zero divisors. Secondly, the authors determine a complete characterization of the functions \(f, g \) whose *-product is slice preserving. Thirdly, the authors classify the pairs of semi-regular functions \( f, g\) defined on \(\Omega\) such that any semi-regular function \(h\) defined on \(\Omega\) which commutes with \(f + g\) and \( f^\ast g\) also commutes with \(f\) and \(g\). The manuscript is well written and includes some background definitions and preliminary connections.
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 7192471
zbMATH Keywords: quaternions
zbMATH Keywords: slice-regular functions
zbMATH Keywords: \(\ast\)-product of slice-regular functions
reviewed by: Linda R. Sons
MaRDI profile type: Publication
full work available at URL: conop-2020-0001
OpenAlex ID: W3004744929
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DOI: 10.1515/CONOP-2020-0001
description: scientific article; zbMATH DE number 7192471