On the spectrum of the Cesàro operator \(C_1\) on \(\overline {bv}_0\cap \ell_{\infty}\) (Q2843883)

Denote the classical spaces of all bounded, convergent, null and absolutely summable sequences by \(\ell_{\infty}\), \(c\), \(c_{0}\) and \(\ell_{1}\), respectively. \(C_{1}\) also denotes the Cesàro operator of order one, as usual. The density of a subset \(E\) of \(\mathbb N\), the set of positive integers, is defined by \(\delta (E)=\lim_{n\to \infty}\frac {1}{n}\sum_{k=1}^{n}\chi_{E}(k)\) provided the limit exists, where \(\chi_{E}\) is the characteristic function of \(E\). A sequence \((x_{k})\) is said to be statistically convergent to \(L\) if for every \(\varepsilon >0\), \(\delta \bigl (\{k\in \mathbb N\!:| x_{k}-L| \geq \varepsilon \}\bigr)=0\). In this case, we write stat-\(\lim x_{k}=L\).NEWLINENEWLINEA sequence \((x_{k})\) is said to be of statistically bounded variation if \((\Delta x_{k})\in \ell_{1}\) such that \(\delta ({k_{i}\in \mathbb N\!:i\in \mathbb N})=1\), where \(\Delta \) denotes the forward difference operator, that is, \(\Delta x_{k_{i}}=x_{k_{i}}-x_{k_{i+1}}\) for all \(i\in \mathbb N\). The class of all statistically bounded variation sequences is denoted by \(\overline {bv}\) and \(\overline {bv}_{0}=\overline {bv}\cap \overline {c}_{0}\), where \(\overline {c}_{0}\) denotes the space of statistically null sequences.NEWLINENEWLINEIn this article, it is proved that \(C_{1}\!\!:\overline {bv}_{0}\cap \ell_{\infty}\to \overline {bv}_{0}\cap \ell_{\infty}\) is a bounded operator with \(\| C_{1}\|_{(\overline {bv}_{0}\cap \ell_{\infty}:\overline {bv}_{0}\cap \ell_{\infty})}=1\) and the spectrum \(\sigma (C_{1},\overline {bv}_{0}\cap \ell_{\infty})\) of the Cesàro operator \(C_{1}\) acting on the set \(\overline {bv}_{0}\cap \ell_{\infty}\) is determined as follows: \(\sigma \bigl (C_{1},\overline {bv}_{0}\cap \ell_{\infty}\bigr)=\{\lambda \in \mathbb C\), \(\bigl | \lambda -\frac {1}{2}\bigr | \leq \frac {1}{2}\}\).