Reduced Embry characterizations of quasinormal operators

We prove that, for any two integers $2\le m<n$, a bounded operator $A$ on a complex Hilbert space satisfying $$A^{\ast k}{A}^k=(A^{\ast }A{)}^k,k\in \{m,n\},$$ is quasinormal. More generally, we show that the inequalities $$A^{\ast m}{A}^m\ge (A^{\ast }A{)}^m\text{and}{A}^{\ast n}{A}^n\le (A^{\ast }A{)}^n$$ already imply quasinormality. Together with the known single-exponent counterexamples, this completely resolves the reduced Embry problem for bounded operators: the identities $A^{\ast k}{A}^k=(A^{\ast }A{)}^k$, $k\in S\subseteq \{1,2,\dots \}$, characterize quasinormality if and only if $S$ contains at least two distinct integers greater than one. The proof combines Hansen’s inequality and the transformer inequality for weighted operator geometric means with an extremal argument for optimal power bounds.