[Paper Review] The spectral theorem for unitary operators based on the $S$-spectrum

Daniel Alpay, Fabrizio Colombo+2 more|arXiv (Cornell University)|Mar 2, 2014

Algebraic and Geometric Analysis19 references4 citations

TL;DR

This paper establishes the spectral theorem for unitary operators on quaternionic Hilbert spaces using the $S$-spectrum, a noncommutative spectral theory framework. By introducing $q$-positive measures and quaternionic spectral measures via the $S$-resolvent operator, it proves that the Riesz projectors and spectral integrals based on the $S$-spectrum recover the classical spectral decomposition, restoring the analogy with complex spectral theory and recovering the matrix case when the $S$-spectrum coincides with the right-spectrum.

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.

Motivation & Objective

• To establish a spectral theorem for unitary operators in the quaternionic setting using the $S$-spectrum, which resolves ambiguity in prior formulations.
• To restore the analogy between quaternionic and complex spectral theory by linking the $S$-spectrum to the Riesz-Dunford functional calculus and spectral measures.
• To define and utilize $q$-positive measures and quaternionic spectral measures based on the $S$-resolvent operator for noncommutative functional calculus.
• To show that the $S$-spectrum generalizes the right-spectrum in finite dimensions and recovers the classical matrix spectral theorem.
• To prove that spectral projectors and integrals based on the $S$-spectrum yield the same decomposition as in the complex case.

Proposed method

• The proof relies on the quaternionic version of Herglotz’s theorem, which uses the notion of $q$-positive measures to represent positive linear functionals.
• Quaternionic spectral measures are constructed via the $S$-resolvent operator and the $S$-spectrum, enabling a noncommutative functional calculus.
• The Cauchy integral formula for slice hyperholomorphic functions is applied to express the $S$-resolvent and Riesz projectors in terms of spectral integrals.
• The spectral projector $\mathcal{P}(\sigma^{0}_{S}(U))$ is computed as a contour integral over the $S$-spectrum, which is shown to equal $E(t_1) - E(t_0)$.
• Fubini’s theorem is used to interchange the order of integration between the spectral measure and the Cauchy kernel, enabling the derivation of the spectral decomposition.
• The key step involves showing that the inner integral $\frac{1}{2\pi}\int_{\partial(\Omega_0 \cap \mathbb{C}_I)} ds_I S_R^{-1}(e^{It}, s)$ equals the characteristic function $\mathbf{1}_{[t_0,t_1]}$, thus linking the $S$-spectrum to spectral projections.

Experimental results

Research questions

• RQ1 How can the spectral theorem for unitary operators be formulated in the quaternionic setting using a consistent spectral theory?
• RQ2 What is the role of the $S$-spectrum in defining spectral measures and projectors for noncommutative operators?
• RQ3 Can the $S$-spectrum recover the classical complex spectral theorem and the matrix case in finite dimensions?
• RQ4 How do $q$-positive measures and the quaternionic Herglotz theorem support the construction of spectral measures?
• RQ5 Is the Riesz projector associated with a spectral set in the $S$-spectrum equal to the difference of spectral projections $E(t_1) - E(t_0)$?

Key findings

• The spectral theorem for unitary operators is established using the $S$-spectrum, ensuring consistency with the complex case.
• The $S$-spectrum is shown to coincide with the right-spectrum in the finite-dimensional case, recovering the classical matrix spectral theorem.
• The Riesz projector $\mathcal{P}(\sigma^{0}_{S}(U))$ is proven to equal $E(t_1) - E(t_0)$, where $E(t)$ is the spectral measure associated with the unitary operator $U$.
• The spectral measure $E(t)$ is constructed via the $S$-resolvent operator and the Cauchy integral formula for slice hyperholomorphic functions.
• The use of $q$-positive measures enables a quaternionic version of Herglotz’s theorem, essential for representing positive functionals in the spectral construction.
• The spectral integral representation $U = \int_0^{2\pi} e^{It} dE_J(t)$ is derived, confirming the spectral decomposition of unitary operators in the $S$-spectrum framework.