## Abstract

In this paper we extend the H^{∞} functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H^{∞} functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H^{∞} functional calculus for (n+1)-tuples of operators applies, in particular, to the Dirac operator.

Original language | English |
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Pages (from-to) | 1544-1584 |

Number of pages | 41 |

Journal | Journal of Functional Analysis |

Volume | 271 |

Issue number | 6 |

DOIs | |

State | Published - 15 Sep 2016 |

Externally published | Yes |

## Keywords

- H functional calculus
- Quaternionic operators
- S-spectrum
- n-tuples of noncommuting operators

## ASJC Scopus subject areas

- Analysis

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