Fine spectra of the discrete generalized Cesàro operator on Banach sequence spaces. (English) Zbl 1443.47005
H. C. Rhaly jun. introduced the discrete generalized Cesàro operator \(C_t\), \(0 \leq t < 1\), in [Proc. Am. Math. Soc. 86, 405–409 (1982; Zbl 0505.47021)] with the following definition \(C_t((x_n)_n):=((1/n+1)\sum_{k=0}^n t^{n-k} x_k)_{n}\) for each \(x=(x_n)_n\) in a sequence space. For \(t=1\), one obtains the classical Cesàro operator. These operators were later investigated by Rhoades and others. This paper utilizes techniques from operator theory and summability to obtain several results about the spectrum and the fine spectrum of the operator \(C_t\), \(0 \leq t < 1\), on \(\ell_{\infty}\) and other Banach sequence spaces related \(\ell_{\infty}\). The spectra depend on the Banach sequence spaces in which the operator acts.
Reviewer: José Bonet (Valencia)
MSC:
| 47A10 | Spectrum, resolvent |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 46B45 | Banach sequence spaces |
Citations:
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