On Sums of Projections

In the paper, for all $n\in\mathbb{N}$, we describe the set $\Sigma_n$ of all real numbers $\alpha$ admitting a collection of projections $P_1,\dots,P_n$ on a Hilbert space $H$ such that $\sum_{k=1}^n P_k=\alpha I$ ($I$ is the identity operator on $H$) and study the problem to find all collections of this kind for a given $\alpha\in\Sigma_n$.