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On the spectrum of the Cesaro operator. (English) Zbl 0548.47017
The Cesàro operator $Cx=y$ where $y\sb n=\frac{x\sb 1+x\sb 2+...+x\sb n}{n}$ is shown to have spectrum $\vert\lambda -{1\over2}\vert\le {1\over2}$ when acting on the space $c\sb 0$ of sequences convergent to zero. C is shown to have no eigenvalues, whilst its adjoint $C\sp*$ has eigenvalues $\vert\lambda -{1\over2}\vert <{1\over2}$ all simple. The methods are similar to those of Halmos et. al. dealing with C on $\ell\sp 2$ (reference given) except that a direct proof of the invertibility of C-$\lambda$ I for $\vert\lambda -{1\over2}\vert >{1\over2}$ is needed.

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