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

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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 nΔn + ... + α1 zΔ+α0 E and L 2= z n a n (z)Δn + ... + 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 n−1 j=s 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.

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Authors
  1. M. I. Nagnybida
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Nagnybida, M.I. On the equivalence of the Euler-Pommier operators in spaces of analytic functions. Ukr Math J 48, 1084–1098 (1996). https://doi.org/10.1007/BF02390966

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  • DOI: https://doi.org/10.1007/BF02390966

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