Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems

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 {e(1), e(2), ... , e(n)} relative to which every element in the given family has the eigenvalue decomposition of the form lambda(1)e(1) + lambda(2)e(2)+ ... + lambda(n)e(n). The simultaneous order spectral decomposition further demands the ordering of eigenvalues lambda(1) >= lambda(2) >= ... >= lambda(n). We characterize this by a pairwise strong operator commutativity condition < x, y > = <lambda(x), lambda(y)> or, equivalently, lambda(x + y) = lambda(x) + lambda(y), where lambda(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.