On the equivalence of the Euler-Pommier operators in spaces of analytic functions - Nagnibida, N. I. (Ukrains’kyi Matematychnyi Zhurnal, 1996)
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.