Projections as averages of isometries on minimal norm ideals

In this paper, we consider projections on minimal norm ideals of B(H)B(H) that are represented as the average of two surjective isometries. We describe projections of the formP(T)=A1TB1+A2TB22,where A1A1, A2A2, B1B1 and B2B2 are unitary operators satisfying a commutativity condition. We also characterize classes of projections of the formP(T)=AT+TtB2andP(T)=ATt+TtB2,with AA and BB unitary operators satisfying a spectral condition.