The operator distances from projections to an idempotent

Abstract: The main purpose of this paper is to give a full characterization of the operator distances from projections to an idempotent, which includes the minimum value, the maximum value and the intermediate values. Let $H$ be a Hilbert space and $\mathbb{B}(H)$ be the set of bounded linear operators on $H$. Given an arbitrary idempotent $Q\in \mathbb{B}(H)$, it is proved that $$|m(Q)-Q|\le |P-Q|\le |I-m(Q)-Q|$$ for every projection $P$ on $H$, in which $I$ is the identity operator on $H$ and $m(Q)$ is a specific projection called the matched projection of $Q$. When $Q\in\mathbb{B}(H)$ is a non-projection idempotent, it is proved that for every number $\alpha$ contained in the interval $\left[|m(Q)-Q|,|I-m(Q)-Q|\right]$, there exists a projection $P\in \mathbb{B}(H)$ such that $|P-Q|=\alpha$. Two uniqueness problems concerning the projections that attain the minimum value or the maximum value are also dealt with.