Totally hereditarily normaloid operators and Weyl’s theorem for an elementary operator

A Hilbert space operator T ∈ B(H) is hereditarily normaloid (notation: T ∈ HN) if every part of T is normaloid. An operator T ∈ HN is totally hereditarily normaloid (notation: T ∈ THN) if every invertible part of T is normaloid. We prove that THN-operators with Bishop’s property (β), also THN-contractions with a compact defect operator such that T −1(0) ⊆ T ∗−1 (0) and non-zero isolated eigenvalues of T are normal, are not supercyclic. Take A and B in THN and let dAB denote either of the elementary operators in B(B(H)): ΔAB and δAB, where ΔAB(X) = AXB −X and δAB(X) = AX −XB. We prove that if non-zero isolated eigenvalues of A and B are normal and B−1(0) ⊆ B∗−1 (0), then dAB is an isoloid operator such that the quasi-nilpotent part H0(dAB −λ) of dAB −λ equals (dAB −λ)−1(0) for every complex number λ which is isolated in σ(dAB). If, additionally, dAB has the single-valued extension property at all points not in the Weyl spectrum of dAB, then dAB, and the conjugate operator d∗AB, satisfy Weyl’s theorem.  2005 Elsevier Inc. All rights reserved