Idempotent operator and its applications in Schur complements on Hilbert C*-module

Abstract The present study proves that T T is an idempotent operator if and only if R ( I − T ∗ ) ⊕ R ( T ) = X {\mathcal{ {\mathcal R} }}\left(I-{T}^{\ast })\oplus {\mathcal{ {\mathcal R} }}\left(T)={\mathcal{X}} and ( T ∗ T ) † = ( T † ) 2 T {\left({T}^{\ast }T)}^{\dagger }={\left({T}^{\dagger })}^{2}T . Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.