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

TL;DR

This paper extends the Taylor formula from classical functional calculus to the non-commutative S-functional calculus for quaternionic operators, introducing a new series expansion for the S-resolvent operator.

Contribution

It generalizes the Taylor formula to the S-functional calculus and introduces a novel series expansion for the S-resolvent operator in non-commutative settings.

Findings

Established a generalized Taylor formula for S-functional calculus.

Developed a new series expansion for the S-resolvent operator.

Overcame non-commutativity obstacles in the proof.

Abstract

The $S$-functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of $n$-tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of $S$-spectrum and of $S$-resolvent operator. Since most of the properties that hold for the Riesz-Dunford functional calculus extend to the S-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 S-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 $S$-resolvent operators associated to the sum of two $n$-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 $S$-resolvent operators. This paper is devoted to researchers working in operators theory and hypercomplex analysis.