On the index of Fredholm pairs of idempotents (Q1034293)

The authors give an alternative proof of the results obtained by \textit{J.\,E.\thinspace Avron, R.\,Seiler} and \textit{B.\,Simon} [J.~Funct.\ Anal.\ 120, No.\,1, 220--237 (1994; Zbl 0822.47033)] related to powers \((P-Q)^{2m+1}\), where \(P\) and \(Q\) are idempotents on a Hilbert space. Namely, they show that, if \((P-Q)^{2n+1}\) is in the trace class, then \((P-Q)^{2m+1}\) also is in the trace class for \(m\geq n\), together with some additional information on \(\text{tr}(P-Q)^{2m+1}\). The authors note that problems of this kind arise in application to the so-called Hall conductance. In their proof, they make use of the technique of block operators.