A simplified formula for the matched projection of an idempotent

Let $\mathcal{L}(H)$ be the set of all adjointable operators on a Hilbert $C^{\ast }$-module $H$. For each $T\in \mathcal{L}(H)$, $T^{\ast }$ denotes its adjoint operator, and $\mathrm{\mid }{T}^{\ast }\mathrm{\mid }$ is the positive square root of $T{T}^{\ast }$. We establish a simplified formula for the matched projection $m(Q)$ of an idempotent $Q\in \mathcal{L}(H)$ as $$m(Q)=\frac{I+\mathrm{\mid }{Q}^{\ast }\mathrm{\mid }-\mathrm{\mid }I-Q^{\ast }\mathrm{\mid }}{2},$$ where $I$ is the identity operator on $H$. This explicit expression allows for the direct derivation of some basic properties of $m(Q)$.