We study global properties of quaternionic slice regular functions (also called extit{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.

s-Regular Functions which Preserve a Complex Slice / Altavilla, A.; DE FABRITIIS, Chiara. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 197:4(2018), pp. 1269-1294. [10.1007/s10231-018-0724-1]

s-Regular Functions which Preserve a Complex Slice

Altavilla, A.;Chiara de Fabritiis
2018-01-01

Abstract

We study global properties of quaternionic slice regular functions (also called extit{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.
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/252488
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