On the equivalence of the Euler-Pommier operators in spaces of analytic functions
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 + ... +...
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| Date: | 1996 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1996
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5269 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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