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Zbl 0831.47020
Okutoyi, J.T.
On the spectrum of $C\sb 1$ as an operator on $bv$.
(English)
[J] Commun. Fac. Sci. Univ. Ankara, Ser. A1 41, No.1-2, 197-207 (1992). ISSN 0251-0871

Summary: In 1985 John Reade determined the spectrum of $C_1$, the Cesàro Operator which is represented by the matrix: $$C_1= \left(\matrix 1 & 0 &&& 0\cdots\\ {1\over 2} &{1\over 2} &&& 0\cdots\\ {1\over 3} & {1\over 3} &&{1\over 3} & 0\cdots\\ &&\cdots\endmatrix\right)$$ regarded as an operator on the space $c_0$ of all null sequences normed by $|x|= \sup_{n\ge 0} |x_n|$.\par It is the purpose of this paper to determine the spectrum of $C_1$ regarded as an operator on the space $bv$ of all sequences $x$ such that $\lim_{n\to \infty} x_n$ exists and $$|x|= \lim_{n\to \infty} |x_n|+ \sum^\infty_{n= 0} |x_{n+ 1}- x_n|< \infty.$$ We do so by proving that $(C_1- \lambda I)^{- 1}\in B(bv)$ for all $\lambda\in C$ such that $|\lambda- {1\over 2}|> {1\over 2}$.
MSC 2000:
*47B37 Operators on sequence spaces, etc.
47A10 Spectrum and resolvent of linear operators
46A45 Sequence spaces

Keywords: Cesàro Operator

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