Sums of compositions of pairs of projections

Abstract

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 n i=1 Q i P i , where P 1 , . . . , Pn, Q 1 , . . . , Qn are orthogonal projections. We show that the smallest number n = n(c) admitting the representation x = n(c) i=1 Q i P i for every x = x * with x ≤ c satisfies 8c + 8 3 ≤ n(c) ≤ 8c + 10.

Around the closures of the set of commutators and the set of differences of idempotent elements of $\mathcal{B}(\mathcal{H})$

“…There exists a substantial literature centred around the characterisation of specific linear and/or multiplicative combinations of projections and idempotents in B(H), and indeed in other C * -algebras [23,10,22,29,11,27,19,30,26,25,24,3,16,1,18].…”