On power series expansions of the SS-resolvent operator and the Taylor formula

The SS-functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of nn-tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of SS-spectrum and of SS-resolvent operator. Since most of the properties that hold for the Riesz–Dunford functional calculus extend to the SS-functional calculus, it can be considered its non commutative version. In this paper we show that the Taylor formula of the Riesz–Dunford functional calculus can be generalized to the SS-functional calculus. The proof is not a trivial extension of the classical case because there are several obstructions due to the non commutativity of the setting in which we work that have to be overcome. To prove the Taylor formula we need to introduce a new series expansion of the SS-resolvent operators associated to the sum of two nn-tuples of operators. This result is a crucial step in the proof of our main results, but it is also of independent interest because it gives a new series expansion for the SS-resolvent operators. This paper is addressed to researchers working in operator theory and in hypercomplex analysis.