The quaternionic spectral theorem has already been considered in the literature, see e.g. , , , however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see , 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 , 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.
D. Alpay, F. Colombo,D. Kimsey and I. Sabadini. The Spectral Theorem for Unitary Operators Based on the S-Spectrum. Milan Journal of Mathematics, vol. 84 (2016), 41-61.