The spectrum and point spectrum of the Cesàro averaging operator $C$ acting on the Fréchet space $C^\infty(\R^+)$ of all $C^\infty$ functions on the interval $[0,\infty)$ 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.
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