An extension theorem for slice monogenic functions and some of its consequences

Abstract. Slice monogenic functions were introduced by the authors in a paper appeared in Israel J. Math. in 2009. The central result of this paper is an extension theorem, which shows that every holomorphic function defined on a suitable domain D of a complex plane can be uniquely extended to a slice monogenic function defined on a domain U_D, determined by D, in a Euclidean space of appropriate dimension. Two important consequences of the result are a structure theorem for the zero set of a slice monogenic function (with a related corollary for polynomials with coefficients in Clifford algebras), and the possibility to construct a multiplicative theory for such functions. Slice monogenic functions have a very important application in the definition of a functional calculus for n-tuples of noncommuting operators.