The H^∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators

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.