The aim of this paper is to study some features of slice semi-regular functions $mathcal{RM}(Omega)$ on a circular domain $Omega$ contained in the skew-symmetric algebra of quaternions $mathbb{H}$ via the analysis of a family of linear operators built from left and right $*$-multiplication on $mathcal{RM}(Omega)$; this class of operators includes the family of Sylvester-type operators $mathcal{S}_{f,g}$. Our strategy is to give a matrix interpretation of these operators as we show that $mathcal{RM}(Omega)$ can be seen as a $4$-dimensional vector space on the field $mathcal{RM}_{mathbb{R}}(Omega)$. We then study the rank of $mathcal{S}_{f,g}$ and describe its kernel and image when it is not invertible. By using these results, we are able to characterize when the functions $f$ and $g$ are either equivalent under $*$-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. We also provide a complete classification of idempotents and zero divisors on product domains of $mathbb{H}$.

Equivalence of slice semi-regular functions via Sylvester operators

Amedeo Altavilla;
2020-01-01

Abstract

The aim of this paper is to study some features of slice semi-regular functions $mathcal{RM}(Omega)$ on a circular domain $Omega$ contained in the skew-symmetric algebra of quaternions $mathbb{H}$ via the analysis of a family of linear operators built from left and right $*$-multiplication on $mathcal{RM}(Omega)$; this class of operators includes the family of Sylvester-type operators $mathcal{S}_{f,g}$. Our strategy is to give a matrix interpretation of these operators as we show that $mathcal{RM}(Omega)$ can be seen as a $4$-dimensional vector space on the field $mathcal{RM}_{mathbb{R}}(Omega)$. We then study the rank of $mathcal{S}_{f,g}$ and describe its kernel and image when it is not invertible. By using these results, we are able to characterize when the functions $f$ and $g$ are either equivalent under $*$-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. We also provide a complete classification of idempotents and zero divisors on product domains of $mathbb{H}$.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/265188
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 8
  • ???jsp.display-item.citation.isi??? ND
social impact