The index of a pair of projections
The authors discuss pairs of self-adjoint projections with \(P- Q\in {\mathcal T}_{2n+ 1}\), the trace ideal, and prove that for \(m\geq n\), \[ \text{tr}(P- Q)^{2m+ 1}= \text{tr}(P- Q)^{2n+ 1}= \dim(\text{Ker } Q\cap \text{Ran } P)- \dim(\text{Ker } P\cap \text{Ran } Q) \] is an integer. They also prove that there exists a unitary \(V\) interchanging \(P\) and \(Q\) if and only if this integer is 0.
scientific article; zbMATH DE number 599757