s-Regular functions which preserve a complex slice

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