On the spectra of the operator $\boldsymbol{B}\left(\widetilde{\boldsymbol{r}},\widetilde{\boldsymbol{s}}\right)$ mapping in $\left(\boldsymbol{w}_{\boldsymbol{\infty}}\left(\boldsymbol{\lambda} \right)\right)_{\boldsymbol{a}}$ and $\left(\boldsymbol{w}_{\boldsymbol{0}}\left(\boldsymbol{\lambda}\right)\right)_{\boldsymbol{a}}$ where $\boldsymbol{\lambda}$ is a nondecreasing exponentially bounded sequence

B. de Malafosse, E. Malkowsky

Given any sequence $a=(a_{n})_{n\geq 1}$ of positive real numbers and any set $E$ of complex sequences, we write $E_{a}$ for the set of all sequences $x=(x_{n})_{n\geq 1}$ such that $x/a=(x_{n}/a_{n})_{n\geq 1}\in E$. We denote by $W_{a}\left( \lambda \right) $ $=\left( w_{\infty }\left( \lambda \right) \right) _{a}$ and $W_{a}^{0}\left( \lambda \right) =\left( w_{0}\left( \lambda \right) \right) _{a}$ the sets of all sequences $x$ such that $\sup_{n}\left( \lambda _{n}^{-1}\sum_{k=1}^{n}\left\vert x_{k}\right\vert /a_{k}\right) <\infty $ and $\lim_{n\rightarrow \infty }\left( \lambda _{n}^{-1}\sum_{k=1}^{n}\left\vert x_{k}\right\vert /a_{k}\right) =0$, where $\lambda $ is a nondecreasing exponentially bounded sequence. In this paper we recall some properties of the Banach algebras $\left( W_{a}\left( \lambda \right) ,W_{a}\left( \lambda \right) \right) $, and $\left( W_{a}^{0}\left( \lambda \right) ,W_{a}^{0}\left( \lambda \right) \right) $, where $a$ is a positive sequence. We then consider the operator $\Delta _{\rho }$, defined by $\left[ \Delta _{\rho }x\right] _{n}=x_{n}-\rho _{n-1}x_{n-1}$ for all $n\geq 1$ with the convention $x_{0}$, $\rho _{0}=0$, and we give necessary and sufficient conditions for the operator $\Delta _{\rho }:E\rightarrow E$ to be bijective, for $E=w_{0}\left( \lambda \right) $, or $w_{\infty }\left( \lambda \right) $. Then we consider the generalized operator of the first difference $B\left( \widetilde{r},\widetilde{s}\right) $, where $\widetilde{r% },\widetilde{s}$ are two convergent sequences, and defined by $\left[B\left( \widetilde{r},\widetilde{s}\right) x\right]_{n}=r_{n}x_{n}+s_{n-1}x_{n-1}$ for all $n\geq 1$ with the convention $x_{0},s_{0}=0$. Then we deal with the operator $B\left( \widetilde{r}, \widetilde{s}\right) $ mapping in either of the sets $W_{a}\left( \lambda \right) $, or $W_{a}^{0}\left( \lambda \right) $. We then apply the previous results to explicitly calculate the spectrum of $B\left( \widetilde{r}, \widetilde{s}\right) $ over each of the spaces $E_{a}$, where $E=w_{0}\left(\lambda \right) $, or $w_{\infty }\left( \lambda \right) $. Finally we give a characterization of the identity $\left( W_{a}\left( \lambda \right) \right) _{B\left( r,s\right) }=W_{b}\left( \lambda \right)$.