An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A∗|2p⩽|A|2p; an invertible operator A∈B(H) is log-hyponormal, A∈(ℓ-H), if log(TT∗)⩽log(T∗T). Let dAB=δAB or ▵AB, where δAB∈B(B(H)) is the generalised derivation δAB(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, , the ascent of (dAB-λ)⩽1, and dAB satisfies the range-kernel orthogonality inequality ‖X‖⩽‖X-(dAB-λ)Y‖ for all X∈(dAB-λ)-1(0) and Y∈B(H). Furthermore, isolated points of σ(dAB) are simple poles of the resolvent of dAB. A version of the elementary operator E(X)=A1XA2-B1XB2 and perturbations of dAB by quasi–nilpotent operators are considered, and Weyl’s theorem is proved for dAB.