S-regular functions which preserve a complex slice

TL;DR

This paper explores properties of quaternionic slice regular functions, introducing new techniques to characterize one-slice preserving functions, and analyzing how various operations affect the preservation of complex slices.

Contribution

It provides new characterizations of one-slice preserving functions and develops factorization theorems for slice regular functions, advancing understanding of their structural properties.

Findings

Characterization of one-slice preserving functions via projectivization.

Introduction of a Hermitian product to express the $*$-product.

Conditions for sum, $*$-product, and conjugation to preserve complex slices.

Abstract

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian" product on slice regular functions which gives us the possibility to express the $*$-product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from $f$ and $g$. Afterwards we are able to determine, under different assumptions, when the sum, the $*$-product and the $*$-conjugation of two slice regular functions preserve a complex slice. We also study when the $*$-power of a slice regular function has this property or when it preserves all complex slices. To obtain these results we prove two factorization theorems: in the first one, we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other which is never-vanishing; in the other one we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one.