Sums of compositions of pairs of projections

TL;DR

This paper establishes necessary and sufficient conditions for representing Hermitian operators as sums of compositions of pairs of projections in infinite-dimensional Hilbert spaces, providing bounds on the minimal number of terms needed.

Contribution

It offers new criteria for such representations and bounds on the minimal number of summands, partially answering a question posed in 2010.

Findings

Characterization of when Hermitian operators can be expressed as sums of compositions of projections.

Bounds on the minimal number of terms needed for the representation.

Partial resolution of a question posed by L. W. Marcoux in 2010.

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.