On sequence spaces equations of the form ET + Fx = Fb for some triangle T

Abstract EN

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 2 E ; in particular, sa(c) denotes the set of all sequences y such that y / a converges.

We denote by w ∞ and w 0 the sets of all sequences y such that supn (n-1 Σ n k = 1 |yk|) =0.

We also use the sets of analytic and entire sequences denoted by ⋀ and Γ and defined by supn |yn| 1 / n < ∞ and limn →∞ |yn|1 / n = 0, respectively.

In this paper we explicitly calculate the solutions of (SSE) of the form ET + Fx = Fb in each of the cases E = c, l ∞, l p (p ≥ 1), wo, Γ or Λ, F = c, or l∞, and T is either of the triangles ∆, or Σ, where ∆ is the operator of the first difference, and Σ is the operator defined by Σ ny = Σ n k = 1 yk.

For instance the solvability of the (SSE) Γ Σ + Λ I = Λ b consists in determining the set of all positive sequences I = (In)n that satisfy the statement : supn {(|yn| / b n) 1 / n} < ∞ if and only if there are u, v ∈ ω with y = u + v such that n lim n → ∞ |Σ uk| k = 11 / n = 0 and sup { (|vn| / In) 1 / n} < ∞ for all y.