Hereditarily normaloid operators

A Banach space operator B() is said to be hereditarily normaloid, HN, if every part of is normaloid; HN is totally hereditarily normaloid, HN, if every invertible part of is also normaloid; and CHN if either HN or ―¸λI is in HN for every complex number ¸. Class CHN is large; it contains a number of the commonly considered classes of operators. We study operators CHN, and prove that the Riesz projection associated with a λ¸ isoσ( ), CHN ∩\ B(H) for some Hilbert space H, is self-adjoint if and only if (―λ I ¯¹(0) ⊆ ( *― λ I)¯¹(0). Operators CHN have the important property that both and the conjugate operator * have the single-valued extension property at points ¸ which are not in the Weyl spectrum of ; we exploit this property to prove a-Browder and a-Weyl theorems for operators CHN.peerReviewe