Universal Decomposition Equalities for Operator Matrices in a Hilbert Space

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This paper establishes some universal decomposition equalities for operator matrices in a Hilbert space. It includes two basic universal operator matrix decompositions for two-by-two and four-by-four operator matrices, and two four-by-four universal operator matrix decompositions for a four-term linear combination $$x_0I + x_1P + x_2Q + x_3PQ$$ , where P and Q are two commutative involutory or two commutative idempotent operators, and $$x_0, \, x_1, \, x_2, \, x_3$$ are four complex scalars. Many consequences are also presented concerning disjoint decomposition equalities, inverses, generalized inverses, collections of involutory, idempotent and tripotent operators generated from these linear combinations, etc.