Hereditarily normaloid operators

Abstract

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