Some results on n-power operators

It is a well known result that 1-power normal operator is equivalent to normal operators. This paper will discuss some results on n-power normal operators on the Hilbert space. It is shown that if T is a diagonal matrix, then T is n-power normal for any integer n>2. We show that the product of two commuting 2- power normal operators is 2-power normal if each operator commutes with the adjoint of the other operator.