Abstract
The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [32], [33], 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 a 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.
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. D. P. Kimsey gratefully acknowledges the support of a Kreitman postdoctoral fellowship. F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written.
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Alpay, D., Colombo, F., Kimsey, D.P. et al. The Spectral Theorem for Unitary Operators Based on the S-Spectrum. Milan J. Math. 84, 41–61 (2016). https://doi.org/10.1007/s00032-015-0249-7
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DOI: https://doi.org/10.1007/s00032-015-0249-7