For a Banach space operator T ∈ B(X), it is proved that if either T is an algebraically, totally hereditarily normaloid operator and the Banach space X is separable, or T satisfies the property that its quasinilpotent part H 0 (T − λ) = (T − λ) −p (0) for all complex numbers λ and some integer p ≥ 1, then f (T ) satisfies generalized Weyl's theorem for every non-constant function f that is analytic on an open neighborhood of σ(T ).