Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems

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