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

Abstract

Abstract. 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.

Schatten class and Berezin transform of quaternionic linear operators

“…The notion of S-spectrum for quaternionic operators arises naturally in the slice hyperholomorphic functional calculus, called Sfunctional calculus or quaternionic functional calculus, which is the quaternionic analogue of the Riesz-Dunford functional calculus for complex operators on a complex Banach space. Recently, it turned out that also the spectral theorem for quaternionic operators (bounded or unbounded) is based on the S-spectrum, see [17,18].…”

“…Moreover, the Fueter mapping theorem revealed a deep relation between slice hyperholomorphic and monogenic functions, which allowed to defined the ℱ‐functional calculus , a functional calculus for monogenic functions that is also based on the S‐spectrum. The spectral theorem for quaternionic bounded or unbounded normal operators based on the S‐spectrum has been proved recently in the papers . The case of compact operators can be found in [19].…”

“…The literature contains several attempts to prove the quaternionic version of the spectral theorem, but the notion of spectrum in the general case was not made clear . The fact that the S ‐eigenvalues and the right eigenvalues are equal explains why it was possible to obtain several results in quaternionic quantum mechanic without the notion of S ‐spectrum.…”

“…The S-resolvent set is defined asThe notion of S-spectrum for quaternionic operators arises naturally in the slice hyperholomorphic functional calculus, called Sfunctional calculus or quaternionic functional calculus, which is the quaternionic analogue of the Riesz-Dunford functional calculus for complex operators on a complex Banach space. Recently, it turned out that also the spectral theorem for quaternionic operators (bounded or unbounded) is based on the S-spectrum, see [17,18].This fact restores for the quaternionic setting the well-known fact that both the Riesz-Dunford functional calculus and the spectral theorem are based on the same notion of spectrum.This fact was not clear for a long time because the literature related to quaternionic linear operators used the notion of right spectrum of a quaternionic operator, which, for a right linear operator T, is defined asThis definition gives rise only to the eigenvalues of T. The other possibility is to define the left spectrum L .T/, where we replace Tv D vs by Tv D sv. In the case of the right eigenvalues, we have a nonlinear operator associated with the spectral problem.…”

BMO- and VMO-spaces of slice hyperholomorphic functions

“…We therefore find that the condition of uniformity in (3.3) cannot be relaxed without losing the validity of Section 3.10. Let 4 denote the Lebesgue measure on ℍ ≅ ℝ 4 , let ∈ 1 ( , 4 ) with ( ) > 0 almost everywhere and let ( ) = ( ) 4 ( ) be the measure with density with respect to4 . If we write the vectorial part of the variable in spherical coordinates, for any function ∈ 1 ( , ) we have…”