Sums of compositions of pairs of projections

Abstract: We give some necessary and sufficient conditions for the possibility to represent a Hermitian operator on an infinite-dimensional Hilbert space (real or complex) in the form $\sum_{i=1}^nQ_iP_i$, where $P_1,\dots,P_n$, $Q_1,\dots,Q_n$ are orthogonal projections. We show that the smallest number $n=n(c)$ admitting the representation $x=\sum_{i=1}^{n(c)}Q_iP_i$ for every $x=x^*$ with $|x|\leq c$ satisfies $8c+\frac83\leq n(c)\leq 8c+10$. This is a partial answer to the question asked by L. W. Marcoux in 2010.