  The quaternionic spectral theorem has already been considered in the
literature, see e.g. [22], [31], [32], however, except for the finite
dimensional case in which the notion of spectrum is associated to an eigenvalue
problem, see [21], it is not specified which notion of spectrum underlies the
theorem. In this paper we prove the quaternionic spectral theorem for unitary
operators using the $S$-spectrum. In the case of quaternionic matrices, the
$S$-spectrum coincides with the right-spectrum and so our result recovers the
well known theorem for matrices. The notion of $S$-spectrum is relatively new,
see [17], and has been used for quaternionic linear operators, as well as for
$n$-tuples of not necessarily commuting operators, to define and study a
noncommutative versions of the Riesz-Dunford functional calculus. The main
tools to prove the spectral theorem for unitary operators are the quaternionic
version of Herglotz's theorem, which relies on the new notion of $q$-positive
measure, and quaternionic spectral measures, which are related to the
quaternionic Riesz projectors defined by means of the $S$-resolvent operator
and the $S$-spectrum. The results in this paper restore the analogy with the
complex case in which the classical notion of spectrum appears in the
Riesz-Dunford functional calculus as well as in the spectral theorem.
