# Journal of Operator Theory

Volume 74, Issue 2, Fall 2015  pp. 307-317.

Sums of compositions of pairs of projections

Authors:  Andrzej Komisarski (1) and Adam Paszkiewicz (2)
Author institution: (1) Department of Probability Theory and Statistics, Faculty of Mathematics and Computer Science, University of Lodz, ul. Banacha 22, 90-238 Lodz, Poland
(2) Department of Probability Theory and Statistics, Faculty of Mathematics and Computer Science, University of Lodz, ul. Banacha 22, 90-238 Lodz, Poland

Summary:  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 $\sum\limits_{i=1}^nQ_iP_i$, 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 $x=\sum\limits_{i=1}^{n(c)}Q_iP_i$ for every $x=x^*$ with $\|x\|\leqslant c$ satisfies $8c+\frac{8}{3}\leqslant n(c)\leqslant 8c+10$. This is a partial answer to the question asked by L.W.~Marcoux in 2010.

DOI: http://dx.doi.org/10.7900/jot.2014jun17.2056
Keywords:  Hilbert space, Hermitian operator, orthogonal projection, composition of orthogonal projections, representation