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Stability in interpolation of families of Banach spaces


Authors: Wei Cao and Yoram Sagher
Journal: Proc. Amer. Math. Soc. 112 (1991), 91-100
MSC: Primary 46M35; Secondary 46B20, 46E99, 47A53
DOI: https://doi.org/10.1090/S0002-9939-1991-1031449-9
MathSciNet review: 1031449
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Abstract: Let $ D$ be a simply connected domain in the complex plane whose boundary $ \Gamma $ is a rectifiable simple closed curve. Let $ \left\{ {A(\gamma )/\gamma \in \Gamma } \right\}$ and $ \left\{ {B(\gamma )/\gamma \in \Gamma } \right\}$ be interpolation families of Banach spaces. Let $ T$ be a linear operator mapping $ A(\gamma )$ continuously into $ B(\gamma )$. For $ z \in D$ let $ {T_z}$ be the restriction of $ T$ to the interpolation space $ {A_z}$. Then $ \{ z \in D/\operatorname{cod}(T_z) = d < \infty$ and $ \dim \operatorname{Ker}({T_z}) = 0 \} $ and $ \{ z \in D/\dim \operatorname{Ker}(T_z) = d < \infty $ and $ T_z$ is onto $ B_z\}$ are open sets.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1031449-9
Article copyright: © Copyright 1991 American Mathematical Society

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