On the solvability of the (SSIE) with operator $${D}_{x} {\mathbf { * }}\left( { s}_{ R}^{{\textbf{0}}} \right) _{{{{\Sigma }} - {\lambda I}}} {{ \subset }}{ s}_{ R}^{{{0}}} $$ , involving the fine spectrum of an operator

Abstract

Given any sequence \(a=(a_{n})_{n\ge 1}\) of positive real numbers and any set E of complex sequences, we write \(E_{a}\) for the set of all sequences \(y=(y_{n})_{n\ge 1}\) such that \(y/a=(y_{n}/a_{n})_{n\ge 1}\in E\); in particular, \(c_{a}\) denotes the set of all sequences y such that y/a converges. In this paper, we use the sum operator \(\Sigma \), defined by \(\Sigma _{n}y=\sum _{k=1}^{n}y_{k}\) for all sequences y, and we determine its spectrum over each of the sets \(s_{a}=(\ell _{\infty })_{a}\) and \(s_{a}^{0}=(c_{0})_{a}\). Then we determine the point, residual and continuous spectra of the operator \(D_{1/R}\Sigma D_{R}\), with \(R>1\), and we solve the special sequence spaces inclusion equations (SSIE), (which are determined by an inclusion, for which each term is a sum or a sum of products of sets of the form \((E_{a})_{\mathcal {T}}\) and \(( E_{f(x)})_{\mathcal {T}}\) where f maps \(U^{+}\) to itself, E is any linear space of sequences and \(\mathcal {T}\) is a triangle) \(D_{x}*(s_{R}^{0})_{\Sigma -\lambda I}\subset s_{R}^{0}\), using the fine spectrum of this operator. The solvability of this (SSIE) consists in determining, for each \(\lambda \in \mathbb {C}\), the set of all sequences \(x\in \omega \) that satisfy the next statement. For every \(y\in \omega \), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{R^{n}}\left( \sum _{k=1}^{n-1}y_{k}-\lambda y_{n}\right) =0\Longrightarrow \lim _{n\rightarrow \infty }x_{n}\left( \frac{y_{n}}{R^{n}}\right) =0\text {.} \end{aligned}$$

Then, we solve this (SSIE) for \(R=1\). Finally, we solve each (SSIE) \(D_{x}*( E_{R})_{\Sigma -\lambda I}\subset s_{R}\), where E is successively equal to \(c_{0}\), c, and \(\ell _{\infty }\).