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 addressed to researchers working in operator theory and in hypercomplex analysis.

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

#### 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 addressed to researchers working in operator theory and in hypercomplex analysis.
##### Scheda breve Scheda completa Scheda completa (DC)
2016
New series expansion for the S-resolvent operators; S-functional calculus for n-tuples of non commuting operators; S-spectrum; Taylor formula for the S-functional calculus; Mathematical Physics; Physics and Astronomy (all); Geometry and Topology
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11311/1007255`
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