Abstract
Let (E 0,E 1) and (H 0,H 1) be two pairs of complex Banach spaces densely and continuously embedded into each other, E 1 ↪ E 0 and H 1 ↪ H 0 and also let \(\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } \). By E θ = [E 0, E 1]θ and H θ = [H 0, H 1]θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by B θ(0,R) we denote an open ball of radius R in the space E θ. Let Φ: B 0(0,R) → H 0 be an analytic mapping taking B 1(0,R) into H 1, and let the estimates
hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball B θ(0,r) of radius r ∈ (0,R) in the space E θ into H θ and
.
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Original Russian Text © A. M. Savchuk, A. A. Shkalikov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 4, pp. 578–581.
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Savchuk, A.M., Shkalikov, A.A. On the interpolation of analytic mappings. Math Notes 94, 547–550 (2013). https://doi.org/10.1134/S0001434613090241
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DOI: https://doi.org/10.1134/S0001434613090241