ON THE OPERATOR EQUATION AB = zBA

In this paper, we study the operator equation AB = zBA for bounded operators A,B on a complex Hilbert space. In (10), 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. 1