SUMS OF COMPOSITIONS OF PAIRS OF PROJECTIONS

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 Sigma(n)(i=1) Q(i)P(i), where P-1, ... , P-n, Q(1), ... , Q(n) are orthogonal projections. We show that the smallest number n = n (c) admiting the representation x = Sigma(n(c))(i=1) Q(i)P(i) for every x = x* with parallel to x parallel to <= c satisfies 8c + 8/3 <= n(c) <= 8c + 10. This is a partial answer to the question asked by L.W. Marcoux in 2010.