A simplified formula for the matched projection of an idempotent

Abstract

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