Relating moments of self-adjoint polynomials in two orthogonal projections

Abstract

Given two orthogonal projections \(\{P,Q\}\) in a non commutative tracial probability space, we prove relations between the moments of \(P+Q\), of the commutator \(i(PQ-QP)\) and of \(P+QPQ\), and those of the angle operator PQP. Our proofs are purely algebraic and enumerative and does not assume P, Q satisfying Voiculescu’s freeness property or being in general position. As far as the sum and the commutator are concerned, the obtained relations follow from binomial-type formulas satisfied by the orthogonal symmetries associated to P and Q together with the trace property. In this respect, they extend those corresponding to the cases where one of the two projections is rotated by a free Haar unitary operator or more generally by a free unitary Brownian motion. As to the operator \(P+QPQ\), we derive autonomous recurrence relations for the coefficients (double sequence) of the expansion of its moments as linear combinations of those of PQP and determine explicitly few of them. These relations are obtained after a careful analysis of the structure of words in the alphabet \(\{P, QPQ\}\). We close the paper by exploring the connection of our previous results to the so-called Kato’s dual pair. Doing so leads to new identities satisfied by their moments and shows also that the interference operator \(J:=PQ+QP-2QPQ\) and the commutator of P and Q have the same moment sequence.