Applications of the Sylvester operator in the space of slice semi-regular functions
Let \(\Omega\) be an axially symmetric domain contained in the skew algebra \(\mathbf{H}\) of quaternions. Let \(\mathcal{R}(\Omega)\) denote the family of slice regular functions on \(\Omega\). A \(\ast\)-product is introduced for functions in \(\mathcal{R}(\Omega)\) whereby along with pointwise sums for addition \(\mathcal{R}(\Omega)\) becomes an associative algebra. Using these operations the authors study the behavior of functions in \(\mathcal{R}\mathcal{M}(\Omega)\) the set of slice semi-regular functions. Heavily relying on results of the authors in [``Equivalence of slice semi-regular functions via Sylvester operators'', Preprint, \url{arXiv:1907.07385}], three main results emerge along with several corollaries. First the authors show for \( f,g\) in \(\mathcal{R}\mathcal{M}(\Omega)\backslash \{0\}\) the kernel of the linear operator \(\mathcal{L}_{f,g}(\mathcal{X})=f^*\mathcal{X}^\ast{g}\) has dimension 2 if and only if exactly one between \( f\) and \( g\) is a zero divisor, while it has dimension 3 if and only if \(f\) and \(g\) are both zero divisors. Secondly, the authors determine a complete characterization of the functions \(f, g \) whose *-product is slice preserving. Thirdly, the authors classify the pairs of semi-regular functions \( f, g\) defined on \(\Omega\) such that any semi-regular function \(h\) defined on \(\Omega\) which commutes with \(f + g\) and \( f^\ast g\) also commutes with \(f\) and \(g\). The manuscript is well written and includes some background definitions and preliminary connections.
\(\ast\)-product of slice-regular functions
scientific article; zbMATH DE number 7192471