A new functional calculus for non commuting operators

Abstract. In this paper we use the notion of slice monogenic functions introduced in a paper by Colombo, Sabadini and Struppa (see Israel J. Math. 2009) to define a new functional calculus for an n-tuple T=(T_1,..., T_n) of not necessarily commuting operators. This calculus is different from the one discussed in the book by Jefferies, Springer Verlag 2004, and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.