Riesz projections for a class of Hilbert space operators

A Hilbert space operator T, T∈B(H), is totally hereditarily normaloid, T∈THN, if every part and (also) every invertible part of T is normaloid. The class THN is large. It is proved that if a T∈THN is such that the isolated eigenvalues of T are normal, then the Riesz projection Pλ associated with a λ ∈ iso σ(T) is self-adjoint and PλH=(T-λ)-1(0)=(T∗-λ¯)-1(0).