Equivalence of slice semi-regular functions via Sylvester operators

Abstract

The aim of this paper is to study some features of slice semi-regular functions $\mathcal{R}\mathcal{M}(\Omega )$ 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 $\ast$-multiplication on $\mathcal{R}\mathcal{M}(\Omega )$; 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 $\mathcal{R}\mathcal{M}(\Omega )$ can be seen as a $4$-dimensional vector space on the field ${\mathcal{R}\mathcal{M}}_{\mathbb{R}}(\Omega )$. 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 $\ast$-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 $\mathbb{H}$.

arXiv commentfinal accepted version

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Cited by in corpus (3)

• $\ast$-Logarithm for Slice Regular Functions
• Slice regular functions as covering maps and global $\star$-roots
• Spherical coefficients of slice regular functions