  In this paper we extend the $H^\infty$ 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 $H^\infty$ functional
calculus based on the notion of $S$-spectrum for both quaternionic operators
and for $n$-tuples of noncommuting operators. We remark that the $H^\infty$
functional calculus for $(n+1)$-tuples of operators applies, in particular, to
the Dirac operator.
