Matrix Transformations and Application to Perturbed Problems of Some Sequence Spaces Equations with Operators

Abstract

Given any sequence z = (zn)n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E; in particular, cz = s(c) z denotes the set of all sequences y such that y/z converges. Starting with the equation Fx = Fb we deal with some perturbed equation of the form E+ Fx = Fb, where E is a linear space of sequences. In this way we solve the previous equation where E =(Ea)T and (E, F) ∈ {(ℓ∞, c) , (c0, ℓ∞) , (c0, c) , (ℓp, c) , (ℓp, ℓ∞) , (w0, ℓ∞)} with p ≥ 1, and T is a triangle.