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On spectrum of the Cesaro operator. (English) Zbl 1090.47019

Let \(T\) be a bounded linear operator on a Banach space. By \(\sigma(T)\), \(\sigma_p(T)\), \(\sigma_c(T)\) and \(\sigma_r(T)\) denote, respectively, the spectrum of \(T\), the point spectrum of \(T\), the continuous spectrum of \(T\) and the residual spectrum of \(T\). For \(1< p<\infty\), consider the Cesàro operator \(C_1\) acting on the space of all \(p\)-absolutely convergent series and let \(1/p+ 1/q= 1\). The main result of this paper says that:
(i) \(\sigma(C_1)= \{\lambda:|\lambda- {q\over 2}|\leq q/2\}\), (ii) \(\sigma_p(C_1)= \phi\), (iii) \(\sigma_c(C_1)= \{\lambda:|\lambda- {q\over 2}|= {q\over 2}\}\), (iv) \(\sigma_r(C_1)= \{\lambda\mid|\lambda- {q\over 2}|< {q\over 2}\}\).
Also, the fine spectrum of the Cesàro operator in the sequence space \(c_0\) is determined.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
46A45 Sequence spaces (including Köthe sequence spaces)
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