TY - JOUR

T1 - The Spectral Theorem for Unitary Operators Based on the S-Spectrum

AU - Alpay, Daniel

AU - Colombo, Fabrizio

AU - Kimsey, David P.

AU - Sabadini, Irene

N1 - Publisher Copyright:
© 2015, Springer International Publishing.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

KW - 35P05

KW - 47B32

KW - 47S10

UR - http://www.scopus.com/inward/record.url?scp=84949525304&partnerID=8YFLogxK

U2 - 10.1007/s00032-015-0249-7

DO - 10.1007/s00032-015-0249-7

M3 - Article

AN - SCOPUS:84949525304

SN - 1424-9286

VL - 84

SP - 41

EP - 61

JO - Milan Journal of Mathematics

JF - Milan Journal of Mathematics

IS - 1

ER -