Phase shift operator and cyclic evolution in finite dimensional Hilbert space

We address the problem of phase shift operator acting as time evolution operator in Pegg-Barnett formalism. It is argued that standard shift operator is inconsistent with the behaviour of the state vector under cyclic evolution. We consider a generally deformed oscillator algebra at q-root of unity, as it yields the same Pegg-Barnett operator and show that shift operator meets our requirement.