# 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

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