Matrix transformations and sequence spaces equations

Abstract

In this paper we study sequence spaces equations (SSE) with operators, which are determined by an identity whose each term is a sum or a sum of products of sets of the form $\chi _{a}\left( T\right)$ and $\chi _{f\left( x\right) }\left( T\right)$ where $f$ maps $U^{+}$ to itself, $\chi $ is either of the symbols $s$, $s^{0}$, or $s^{\left( c\right) }$. Then we solve five (SSE) of the form $\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }$, where $\chi $, $\chi ^{\prime }$ are either $s^{{{}0}}$, $s^{\left( c\right) }$, or $s$. We apply the previous results to the solvability of the systems $s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}$, $s_{x}\supset s_{b}$ and $s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }$, $s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }$. Finally we solve the (SSE) with operators defined by $\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }$ where $\chi $ is either $s^{0}$, or $s$.