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On the spectrum of the Cesaro operator. (English) Zbl 0548.47017
The Cesàro operator Cx=y where y n =x 1 +x 2 +···+x n n is shown to have spectrum |λ-1 2|1 2 when acting on the space c 0 of sequences convergent to zero. C is shown to have no eigenvalues, whilst its adjoint C * has eigenvalues |λ-1 2|<1 2 all simple. The methods are similar to those of Halmos et. al. dealing with C on 2 (reference given) except that a direct proof of the invertibility of C-λ I for |λ-1 2|>1 2 is needed.

MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10Spectrum and resolvent of linear operators
40G05Cesàro, Euler, Nörlund and Hausdorff methods