Solvability of Sequence Spaces Equations Using Entire and Analytic Sequences and Applications

Given any sequence z = (z_n)n≥1 of positive real numbers and any set E of complex sequences, we write E_z for the set of all sequences y = (y_n)n≥1 such that y/z = (y_n)/z_n))n≥1 ∈ E; in particular, s_z^(c)) denotes the set of all sequences y such that y/z converges. In this paper we deal with special sequence spaces equations (SSE) with operators, which are determined by an identity each term of which is a sum or a sum of products of sets of the form X_a (T) and X_f(x) (T) where U⁺ maps to itself, x is either of the symbols s, s⁰, Γ, or Λ. We solve (SSE) of the form X_a+X_x' = X_b', and systems of the form X_a+X_x'(Δ) = X_b', and X_b' ⊂ X_x', where X, X' are any of the symbols s^o, s^(c), s, Γ, or Λ. For instance the system s_a^(c) + s_x^(c) (Δ) = s_b^(c) and s_b^(c) ⊂ s_x^(c) where Δ is the operator of the first difference means that, b_n/x_n → l₁ (n → ∞), for some l₁ ∈ C, and for any given y ∈ ω, we have y_n/b_n → l₁ (n → ∞ if and only if there are u, v ∈ s such that y = u + v and u_n/a_n → l₁ and Δν_n/x_n → l" (n → ∞) for some scalars l, l' and l".