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 \(T \in \mathcal {B}(\mathcal {H})\) is called of class \([\mathfrak {GN}]\) 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 \([\mathfrak {GN}]\) . 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 \(S, T \in \mathcal {M}_{n}( \mathbb {C}) \) such that \(ST, TS \in [\mathfrak {GN}]\) . Finally, we generalize Fuglede-Putnam commutativity theorem for class \([\mathfrak {GN}]\) of matrices.
