On D-hyponormal operators - Mansour Dana; Fateme Kousari; Ramesh Yousefi (Journal of Inequalities and Applications)
Abstract A Drazin invertible operator T on a Hilbert space is said to be of class [ D H ] $[DH] $ if T ∗ T D ≥ T D T ∗ $T^{*}T^{D}≥ 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 ( β ) $(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 H 0 ( T − λ ) $\mathcal{H}_{0} (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 ) $σ (T) $ and E be the Riesz idempotent for λ. We prove that (1) if λ ≠ 0 $λ ≠ 0 $ , then E is self-adjoint and E H = N ( T − λ ) = N ( T − λ ) ∗ $E\mathcal{H} = \mathcal{N}(T - λ ) = \mathcal{N}{(T - λ )^{*}} $ ; (2) if λ = 0 $λ = 0 $ , then E H = N ( T ) k $E\mathcal{H}= \mathcal{N}(T)^{k} $ .