Abstract
The spectrum and point spectrum of the Cesàro averaging operator \(\mathsf {C}\) acting on the Fréchet space \(C^\infty ({\mathbb R}_+)\) 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 \(\mathsf {C}\) is also investigated.
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Acknowledgments
The research of the first two authors was partially supported by the projects MTM2013-43540-P, GVA Prometeo II/2013/013 and GVA ACOMP/2015/186 (Spain).
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Communicated by A. Constantin.
An erratum to this article can be found at http://dx.doi.org/10.1007/s00605-016-0975-0.
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Albanese, A.A., Bonet, J. & Ricker, W.J. Dynamics and spectrum of the Cesàro operator on \(C^\infty ({\mathbb R}_+)\) . Monatsh Math 181, 267–283 (2016). https://doi.org/10.1007/s00605-015-0863-z
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DOI: https://doi.org/10.1007/s00605-015-0863-z