Abstract. We shall show that the Riesz idempotent E λ of every * -paranormal operator T on a complex Hilbert space H with respect to each isolated point λ of its spectrum σ(T ) is self-adjoint and satisfies E λ H = ker(T − λ) = ker(T − λ) * . Moreover, Weyl's theorem holds for * -paranormal operators and more general for operators T satisfying the norm condition T x n ≤ T n x x n−1 for all x ∈ H. Finally, for this more general class of operators we find a sufficient condition such that E λ H = ker(T − λ) = ker(T − λ) * holds.