On the index of Fredholm pairs of idempotents

Using the technique of block-operators, in this note, we prove that if P and Q are idempotents and (P . Q)2n+1 is in the trace class, then (P . Q)2m+1 is also in the trace class and tr(P . Q)2m+1 = dim(.(P) . .(Q).) . dim(.(P). . .(Q)), for all m >- n. Moreover, we prove that dim(.(P) . .(Q).) = dim(.(P). . .(Q)) if and only if there exists a unitary U such that UP = QU and PU = UQ, where .(T) denotes the range of T.