In this paper, we study the operator equation $AB = zBA$ for 
bounded operators $A, B$ on a complex Hilbert space. In \cite{YD}, J. Yang and 
H.-K. Du proved that if $A$ and $B$ are normal operators, then $|z| = 1$ 
by using the Fuglede-Putnam Theorem. In this paper, we give 
an elementary proof of this result 
without using the Fuglede-Putnam Theorem and some examples.
Then we shall relax normality in the result by Yang and Du.
A quasinormality of an operator is given by using Aluthge transformation and 
the operator equality.