A Drazin invertible operator T on a Hilbert space is said to be of class [DH] if T ∗TD ≥ TDT ∗. 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 (β). 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 H0(T – λ) of T both when T is D-hyponormal and when T is algebraically D-hyponormal. Finally, let λ be an isolated point of σ(T) and E be the Riesz idempotent for λ. We prove that (1) if λ = 0, then E is self-adjoint and EH = N (T – λ) = N (T – λ) ∗; (2) if λ = 0, then EH = N (T) k .
