S-regular functions which preserve a complex slice

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 $\ast$-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 $\ast$-product and the $\ast$-conjugation of two slice regular functions preserve a complex slice. We also study when the $\ast$-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.

arXiv comment23 pages, to appear in Annali di Matematica Pura e Applicata

References in corpus (9)

• Slice regular functions on real alternative algebras
• Continuous slice functional calculus in quaternionic Hilbert spaces
• Twistor transforms of quaternionic functions and orthogonal complex structures
• The C-property for slice regular functions and applications to the Bergman space
• Regular Moebius transformations of the space of quaternions
• Power and spherical series over real alternative *-algebras
• Some properties for quaternionic slice-regular functions on domains without real points
• Approximation by polynomials on quaternionic compact sets
• Twistor interpretation of slice regular functions

Cited by in corpus (5)

• Equivalence of slice semi-regular functions via Sylvester operators
• S--polyregular Bargmann spaces
• Slice regular functions as covering maps and global $\star$-roots
• Implementing zonal harmonics with the Fueter principle
• Spherical coefficients of slice regular functions