Solvability of certain sequence spaces equations with operators

Bruno de Malafosse

In this paper we deal with special sequence space 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 $\chi_a(T)$ and $\chi_{f(x)}(T)$ where $f$ map $U^+$ to itself and $\chi$ is any of the symbols $s$, $s^0$, or $s^{(c)}$. Among other things under some conditions we solve (SSE) with operators $\chi_a(C(\lambda)D_{\tau})+\chi_x(C(\mu)D_{\tau})=\chi_b$, and $\chi_a(C(\lambda)C(\mu))+\chi_x(C(\lambda\sigma)C(\mu))=\chi_b$ where $\chi\in\{s,s^0\}$, and $\chi_a(C(\lambda)D_{\tau})+s_x^0(C(\mu)D_{\tau}) =\chi_b$ where $\chi$ is either of the symbols $s$, or $s^{(c)}$ and $C(\nu)D_{\tau}$ is a factorable matrix.