The Index of a Pair of Projections

We discuss pairs of self-adjoint projections with P − Q ∈ J2n + 1, the trace ideal, and prove that for m ≥ n, tr(P − Q)2m + 1 = tr(P − Q)2n + 1 = dim(Ker Q ∩ Ran P) − dim(Ker P ∩ Ran Q) is an integer. We also prove that there exists a unitary V interchanging P and Q if and only if this integer is 0.