Reduced commutativity of moduli of operators.

In this paper, we investigate the question of when the equations A ⁎ s A s = ( 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 < n , then A is quasinormal. Moreover, if A is invertible and S = { m , n , n + m } with m ⩽ n , then A is normal. The case when S = { m , m + n } and A ⁎ n A n ⩽ ( A ⁎ A ) n is also discussed. [ABSTRACT FROM AUTHOR]