Applications of the Sylvester operator in the space of slice semi-regular functions

In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ââžâ™f,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the-product, giving a complete classification of the cases when the functions fv, gv and fv gv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their ∗-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f ∗ g, we have that h commutes with f and g, too.