Quaternionic Regularity via Analytic Functional Calculus

Abstract

Let ${\mathbb M}$ be the complexification of the quaternionic algebra ${\mathbb H}$. For each function $F:U\mapsto {\mathbb M}$, where $U\subset {\mathbb C}$, we define a transformation $F_{\mathbb H}:U_{\mathbb H}\mapsto {\mathbb M}$, where $U_{\mathbb H}\subset {\mathbb H}$ is associated to U, via an elementary functional calculus, using the spectra of quaternions, and characterize those transformations $F_{\mathbb H}$, which are actually ${\mathbb H}$-valued. In particular, we show that the slice hyperholomorphy can be characterized via a Cauchy type transform, acting on the space of analytic ${\mathbb M}$-valued stem functions.