  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
$\mathbb{H}$.
