An elementary operator with log-hyponormal, p-hyponormal entries Original Research Article

A Hilbert space operator image is p-hyponormal, Aset membership, variant(p-H), if A*2pless-than-or-equals, slantA2p; an invertible operator image is log-hyponormal, Aset membership, variant(ℓ-H), if log(TT*)less-than-or-equals, slantlog(T*T). Let dAB=δAB or up triangle, openAB, where image is the generalised derivation δAB(X)=AX-XB and image is the elementary operator up triangle, openAB(X)=AXB-X. It is proved that if A,B*set membership, variant(ℓ-H)union or logical sum(p-H), then, for all complex λ, image, the ascent of (dAB-λ)less-than-or-equals, slant1, and dAB satisfies the range-kernel orthogonality inequality double vertical barXdouble vertical barless-than-or-equals, slantdouble vertical barX-(dAB-λ)Ydouble vertical bar for all Xset membership, variant(dAB-λ)-1(0) and image. 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.