A Hilbert space operator A ∈ B ( H ) is p-hyponormal, A ∈ ( p - H ) , if | A ∗ | 2 p ⩽ | A | 2 p ; an invertible operator A ∈ B ( H ) is log-hyponormal, A ∈ ( ℓ - H ) , if log ( TT ∗ ) ⩽ log ( T ∗ T ) . Let d AB = δ AB or ▵ AB , where δ AB ∈ B ( B ( H ) ) is the generalised derivation δ AB ( X ) = AX - XB and ▵ AB ∈ B ( B ( H ) ) is the elementary operator ▵ AB ( X ) = AXB - X . It is proved that if A , B ∗ ∈ ( ℓ - H ) ∪ ( p - H ) , then, for all complex λ , ( d AB - λ ) - 1 ( 0 ) ⊆ ( d A ∗ B ∗ - λ ¯ ) - 1 ( 0 ) , the ascent of ( d AB - λ ) ⩽ 1 , and d AB satisfies the range-kernel orthogonality inequality ‖ X ‖ ⩽ ‖ X - ( d AB - λ ) Y ‖ for all X ∈ ( d AB - λ ) - 1 ( 0 ) and Y ∈ B ( H ) . Furthermore, isolated points of σ ( d AB ) are simple poles of the resolvent of d AB . A version of the elementary operator E ( X ) = A 1 XA 2 - B 1 XB 2 and perturbations of d AB by quasi–nilpotent operators are considered, and Weyl’s theorem is proved for d AB .