Solvability of the Sequence Spaces Equations With Operators C<sub>a</sub>+C<sub>x</sub> = (C<sub>b</sub>)<sub>B</sub><sub>(r,s)</sub> And C<sub>a</sub>+C<sub>x</sub> = (C<sub>b</sub>)<sub>G</sub><sub>(α,α)</sub>

Abstract

Given any sequence a = (an)n?1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n?1 such that y/a = (yn/an)n?1 ? E; in particular, ca, (or s(c) a ) denotes the set of all sequences y such that y/a converges. In this paper, we solve sequence spaces equations of the form (Ex)B(r,s) = Ea, where E 2 {c0, c, `1}. Then we apply these results to the solvability of each of the (SSE) with operators ca + cx = (cb)B(r,s) and ca + cx = (cb)G(,), where B (r, s) is a double band matrix, and G(, ) is the factorable matrix with positive sequences and , that is, the triangle whose the nonzero entries are defined by [G(, )]nk = nk.