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 ${\sum }_{i=1}^n{Q}_i{P}_i$, 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 $x={\sum }_{i=1}^{n(c)}{Q}_i{P}_i$ for every $x=x^{\ast }$ with $\Vert x\Vert \le c$ satisfies $8c+\frac{8}{3}\le n(c)\le 8c+10$. This is a partial answer to the question asked by L. W. Marcoux in 2010.