Equivalence of slice semi-regular functions via Sylvester operators

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 $\mathbb{H}$.