Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems

Abstract

This article deals with necessary and sufficient conditions for a family of elements in a Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by a well-known matrix theory result that any family of pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show that in the setting of a general Euclidean Jordan algebra, any family of pairwise operator commuting elements has a simultaneous spectral decomposition, i.e., there exists a common Jordan frame {e₁, e₂, . . . , eₙ} relative to which every element in the given family has the eigenvalue decomposition of the form λ₁e₁ + λ₂e₂ + · · · + λₙeₙ. The simultaneous order spectral decomposition further demands the ordering of eigenvalues λ₁ ≥ λ₂ ≥ · · · ≥ λₙ. We characterize this by pairwise strong operator commutativity condition (x, y) = (λ(x), λ(y)), or equivalently, λ(x + y) = λ(x) + λ(y), where λ(x) denotes the vector of eigenvalues of x written in the decreasing order. Going beyond Euclidean Jordan algebras, we formulate commutativity conditions in the setting of the so-called Fan-Theobald-von Neumann system that includes normal decomposition systems (Eaton triples) and certain systems induced by hyperbolic polynomials.