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In this paper we extend the $H^\infty$ functional calculus to quaternionicoperators and to $n$-tuples of noncommuting operators using the theory of slicehyperholomorphic functions and the associated functional calculus, called$S$-functional calculus. The $S$-functional calculus has two versions one forquaternionic-valued functions and one for Clifford algebra-valued functions andcan be considered the Riesz-Dunford functional calculus based on slicehyperholomorphicity because it shares with it the most important properties.  The $S$-functional calculus is based on the notion of $S$-spectrum which, inthe case of quaternionic normal operators on a Hilbert space, is also thenotion of spectrum that appears in the quaternionic spectral theorem.  The main purpose of this paper is to construct the $H^\infty$ functionalcalculus based on the notion of $S$-spectrum for both quaternionic operatorsand for $n$-tuples of noncommuting operators. We remark that the $H^\infty$functional calculus for $(n+1)$-tuples of operators applies, in particular, tothe Dirac operator.

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