A Drazin invertible operator T on a Hilbert space is said to be of class $[DH] $ if $T^{*}T^{D}\geq T^{D}T^{*} $ . Our findings contribute to the deeper understanding of D-hyponormal operators by proving several key inequalities and generalizing fundamental results. We show that T has the Bishop’s property $(\beta ) $ . We prove the Putnam’s inequality, Berger–Shaw’s inequality and Weyl’s theorem for D-hyponormal operators. Also, we prove a Fuglede–Putnam commutativity theorem for D-hyponormal operators. In the following, we extend Kaplansky’s well-known result on products of normal operators to D-hyponormal operators. Moreover, we characterize the quasinilpotent part $\mathcal{H}_{0} (T- \lambda ) $ of T both when T is D-hyponormal and when T is algebraically D-hyponormal. Finally, let λ be an isolated point of $\sigma (T) $ and E be the Riesz idempotent for λ. We prove that (1) if $\lambda \neq 0 $ , then E is self-adjoint and $E\mathcal{H} = \mathcal{N}(T - \lambda ) = \mathcal{N}{(T - \lambda )^{*}} $ ; (2) if $\lambda = 0 $ , then $E\mathcal{H}= \mathcal{N}(T)^{k} $ .
