Almost Weighted Composition Operators Between Banach Function Algebras

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y , respectively, and ρ, τ : I → A, S, T : I → B be maps on a non-empty set I whose ranges are closed under multiplication and contain exponential functions. In this paper, we first show that if ‖S(p)‖Y = ‖ρ(p)‖X , ‖T (p)‖Y = ‖τ(p)‖X and ‖S(p)T (q)‖Y = ‖ρ(p)τ (q)‖X , for all p, q ∈ I , then there exists a homeomorphism ϕ from Šilov boundary ∂ A of A onto ∂ B such that for each x ∈ ∂ A and p ∈ I , |S(p)(ϕ(x))| = |ρ(p)(x)| and |T (p)(ϕ(x))| = |τ(p)(x)|. Then we prove that, if for some ε ≥ 0, Ranπ (T (p)S(q)) is contained in an ε‖τ(p)ρ(q)‖-neighborhood of Ranπ (τ (p)ρ(q)), for all p, q ∈ I , then, under a certain condition, there exist continuous functions α, β ∈ B such that |α(ϕ(x))T (p)(ϕ(x)) − τ(p)(x)| ≤ 4ε|τ(p)(x)| and |β(ϕ(x))S(p)(ϕ(x))−ρ(p) (x)| ≤ 4ε|ρ(p)(x)|, for all p ∈ I and x ∈ ∂(A). Our results can be applied for Banach algebras of Lipschitz functions.