The Moore–Penrose Inverse and Product Decomposition of Idempotent Operators on Hilbert C*-Modules

We study the Moore–Penrose inverse of idempotent operators on Hilbert C*-modules. First, we extend the computation of the Moore–Penrose inverse of an idempotent operator and its difference from the range projection to this setting. This leads to an explicit formula for the Moore–Penrose inverse of the sum of an idempotent and its adjoint. Furthermore, we establish a decomposition of an idempotent operator into a product of two commuting idempotents and clarify the relationship between their Moore–Penrose inverses and that of the original operator. We also analyze spectral properties and operator norms, obtaining sharp norm bounds.