Equivalence of slice semi-regular functions via Sylvester operators

The aim of this paper is to study some features of slice semi-regular functions SEM(Ω) on a circular domain Ω contained in the skew-symmetric algebra of quaternions H via the analysis of a family of linear operators built from left and right *-multiplication on SEM(Ω); this class of operators includes the family of Sylvester-type operators S_{f,g}. Our goal is achieved by a strategy based on a matrix interpretation of these operators as we show that SEM(Ω) can be seen as a 4-dimensional vector space on the field SEM_{R}(Ω). We then study the rank of S_{f,g} and describe its kernel and image when it is not invertible, finding meaningful differences in the cases when the rank is either 2 or 3. 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. In this way, informations about the operator obtained by linear algebra techniques give as a significant application the solution of a problem in an area of function theory which had an remarkable development in the last decade. We also provide a complete classification of idempotents and zero divisors on product domains of H.