A note on functions which operate

Let 𝐅, B denote two families of functions a, b: X → Y. A function F:Z ⊆Y → Y is said to operate in (𝐅, B) provided that for each a ∈𝐅 with range (a)⊆ Z we have F(a)∈ B. Let G denote a locally compact Abelian group. In this paper we characterize the functions which operate in two cases: (i) 𝐅 = ϕ_r(G) = positive definite functions on G with ϕ(e) = r and B = ϕ_i.d.,.(G) = infinitely divisible positive definite functions on G with ϕ(e) = s. (ii) 𝐅 = B = ϕ^∼(G) = ϕ_i.d.,.(G).