On the equivalence of the Euler-Pommier operators in spaces of analytic functions

Abstract

In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ⊂ ℂ (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L ₁=α _n z ⁿ Δ ⁿ + ... + α₁ zΔ+α₀ E and L ₂= z ⁿ a _n (z)Δ ⁿ + ... + za ₁(z)Δ+a ₀(z)E, where δ: (Δƒ)(z)=(f(z)-ƒ(0))/z is the Pommier operator in A(G), n ∈ ℕ, α _n ∈ ℂ, a _k (z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σ _j=s ⁿ⁻¹ α _j+1 ∈ 0, s=0,1,...,n−1. We also prove that the operators z ^s+1Δ+β(z)E, β(z) ∈ A _R , s ∈ ℕ, and z ^s+1 are equivalent in the spaces A _R, 0šRš-∞, if and only if β(z) = 0.