Some applications of the matched projections of idempotents

For every idempotent $Q$ on a Hilbert space $H$, the matched projection $m(Q)$ is a well-established concept. This paper explores several applications of the matched projections. The first application addresses the distances from projections on $H$ to a given idempotent $Q$. Using $m(Q)$, a complete characterization of these distances is established, covering the minimum, maximum, and intermediate values. The second application focuses on the $C^*$-algebra $C^*\{Q\}$ generated by a single non-projection idempotent $Q$. A new $4\times 4$ block matrix representation of $Q$, induced by $m(Q)$, yields novel formulas for $Q$, leading to a full characterization of all elements in $C^*\{Q\}$ via explicit $4\times 4$ block matrices. Furthermore, for each $r>1$, a family of universal $r$-idempotents is introduced. These idempotents possess a universal property distinct from known properties of projection pairs. Some necessary and sufficient conditions are provided for such universal $r$-idempotents. The third application presents new characterizations of the numerical ranges. An operator version of the elliptical range theorem is established. Using a general non-projection idempotent $Q$ and its matched projection $m(Q)$, a non-quadratic operator is constructed, and its numerical range is described in detail. Additionally, another operator is introduced whose numerical range closure is not an elliptical disk, and the numerical range itself is neither closed nor open.