Application of the infinite matrix theory to the solvability of certain sequence spaces equations with operators

Abstract

In this paper we deal with special {ıt sequence spaces equations (SSE)with operators}, which are determined by an identity whose each term is a{ıt sum or a sum of products of sets of the form ${\chi }_a(T)$ and ${\chi }_{f(x)}(T)$}where $f$ maps $U^{+}$ to itself, and $\chi$ is any of the symbols $s$, $s^0$, or $s^{(c)}$. We solve the equation ${\chi }_x(\mathrm{\Delta })={\chi }_b$ where $\chi$ is any of the symbols $s$, $s^0$,or $s^{(c)}$ and determine the solutions of (SSE) with operatorsof the form $({\chi }_a\ast {\chi }_x+{\chi }_b)(\mathrm{\Delta })={\chi }_{\eta }$ and $[{\chi }_a\ast ({\chi }_x{)}^2+{\chi }_b\ast {\chi }_x](\mathrm{\Delta })={\chi }_{\eta }$and ${\chi }_a+{\chi }_x(\mathrm{\Delta })={\chi }_x$ where $\chi$ isany of the symbols $s$, or $s^0$.