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

Abstract

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 d A B denote either of the elementary operators in B ( B ( H ) ) : Δ A B and δ A B , where Δ A B ( X ) = A X B − X and δ A B ( X ) = A X − X B . We prove that if non-zero isolated eigenvalues of A and B are normal and B −1 ( 0 ) ⊆ B ∗ −1 ( 0 ) , then d A B is an isoloid operator such that the quasi-nilpotent part H 0 ( d A B − λ ) of d A B − λ equals ( d A B − λ ) −1 ( 0 ) for every complex number λ which is isolated in σ ( d A B ) . If, additionally, d A B has the single-valued extension property at all points not in the Weyl spectrum of d A B , then d A B , and the conjugate operator d A B ∗ , satisfy Weyl's theorem.