Reduced commutativity of moduli of operators

In this paper, we investigate the question of when the equations A⁎sAs=(A⁎A)s, s∈S, where S is a finite set of positive integers, imply the quasinormality or normality of A. In particular, it is proved that if S={p,m,m+p,n,n+p}, where p⩾1 and 2⩽m