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, …, e_n} relative to which every element in the given family has the eigenvalue decomposition of the form λ₁e₁ + λ₂e₂ + · · · + λ_ne_n. The simultaneous order spectral decomposition further demands the ordering of eigenvalues λ₁ ≥ λ₂ ≥ · · · ≥ λ_n. We characterize this by a 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.

本文讨论欧几里得约旦代数中的元素族具有同时（阶）谱分解的必要条件和充分条件。受著名的矩阵理论结果的启发，即任何成对交换复数厄尔米特矩阵族都同时（单一地）对角化，我们表明，在一般欧几里得约旦代数的设置中，任何成对算子交换元素族都具有同时谱分解，即存在一个共同的 Jordan 框架 { e ₁ , e ₂ , …, en } 相对于给定族中的每个元素都具有λ ₁ e ₁  +  λ形式_{的特征值分解2} e ₂  + · · · +  λ _n e _n。同时阶谱分解进一步要求对特征值进行排序λ ₁  ≥  λ ₂  ≥ · · · ≥  λ _n。我们通过成对的强算子交换性条件 〈x , y〉 = 〈λ ( x ), λ ( y )〉 或等效地λ ( x  +  y ) =  λ ( x ) +  λ ( y ) 来表征这一点，其中λ ( x ) 表示按降序排列的x的特征值向量。超越欧几里得约旦代数，我们在所谓的 Fan-Theobald-von Neumann 系统的设置中制定了交换条件，该系统包括正常分解系统（伊顿三元组）和由双曲多项式诱导的某些系统。