## Abstract

The $$S$$S-functional calculus is a functional calculus for (Formula presented.)-tuples of not necessarily commuting operators that can be considered a higher-dimensional version of the classical Riesz–Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left (Formula presented.) and the right one (Formula presented.), where (Formula presented.) and (Formula presented.) is an (Formula presented.)-tuple of noncommuting operators. The two S-resolvent operators satisfy the S-resolvent equations (Formula presented.), and (Formula presented.), respectively, where (Formula presented.) denotes the identity operator. These equations allow us to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.

Original language | English |
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Pages (from-to) | 1939-1968 |

Number of pages | 30 |

Journal | Journal of Geometric Analysis |

Volume | 25 |

Issue number | 3 |

DOIs | |

State | Published - 20 Jul 2015 |

## Keywords

- Left s-resolvent operator
- Projectors
- Quaternionic operators
- Resolvent equation
- Right s-resolvent operator
- n-tuples of noncommuting operators
- s-spectrum

## ASJC Scopus subject areas

- Geometry and Topology