In this paper we introduce and analyze a new class of generalized normal operators, namely skew D-quasi-normal operators, for a bounded linear operator T on a Hilbert space H using the Drazin T^D inverse of T. After establishing the basic properties of such operators, we give examples and discuss how this class of operators is distinct from several other operator classes. We also generalize a very famous result on normal operators, due to Fuglede.
