Dynamics and spectrum of the Cesàro operator on C∞(R+)

The spectrum and point spectrum of the Cesàro averaging operator C acting on the Fréchet space C ∞ ( R + ) of all C ∞ functions on the interval [ 0 , ∞ ) are determined. We employ an approach via C 0 -semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of C is also investigated.