In this paper, we seek to investigate this important question: Is it possible that a Drazin invertible operator and its Drazin inverse are not normal, but their sum is normal? Exploring to find the answer to this question made us able to introduce a new class of operators. A Drazin invertible operator T ∈ B(H) is called of class [GN] if T ∗ and T + T D commute or equivalently T + T D is normal. This class contains the class of normal operators. First, we give a list of conditions on an operator T , each of which is equivalent to T being of class[GN]. We also present some basic properties of these operators. Moreover, we obtain the matrix representation of these operators. We generalize a very famous result on normal operators, due to Kaplansky. Furthermore, we investigate a necessary and sufficient condition for S, T ∈ Mn(C) such that ST , T S ∈ [GN]. Finally, we generalize Fuglede-Putnam commutativity theorem for class [GN] of matrices.
