Form and index of Ginsparg-Wilson fermions

We clarify the questions rised by a recent example of a lattice Dirac operator found by Chiu. We show that this operator belongs to a class based on the Cayley transformation and that this class on the finite lattice generally does not admit a nonvanishing index, while in the continuum limit, due to operator properties in Hilbert space, this defect is no longer there. Analogous observations are made for the chiral anomaly. We also elaborate on various aspects of the underlying sum rule for the index.