The Spectral Theorem for Unitary Operators Based on the S-Spectrum
By Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini. Milan Journal of Mathematics, 2014
52 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics
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In Applied Mathematics
- Irene Sabadini
- Fabrizio Colombo
- Daniele C. Struppa
- Caterina Stoppato
- Graziano Gentili
- Riccardo Ghiloni
- A. Perotti
- Daniel Alpay
Citations in the corpus
- Entire slice regular functions (2015, methodology): “The spectral theorem based on the S-spectrum for bounded and for unbounded quaternionic normal operators on a Hilbert space was developed in [10, 11, 115].”