Abstract
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.
