OPERATORS THAT ARE NUCLEAR WHENEVER THEY ARE NUCLEAR FOR A LARGER RANGE SPACE

Let $X$ be a Banach space and let $Y$ be a closed subspace of a Banach space $Z$. The following theorem is proved. Assume that $X^*$ or $Z^*$ has the approximation property. If there exists a bounded linear extension operator from $Y^*$ to $Z^*$, then any bounded linear operator $T:X\rightarrow Y$ i...

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Published in:Proceedings of the Edinburgh Mathematical Society 2004-10, Vol.47 (3), p.679-694
Main Author: Oja, Eve
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Language:English
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approximation property
extension operator
nuclear operators
projective tensor products
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description Let $X$ be a Banach space and let $Y$ be a closed subspace of a Banach space $Z$. The following theorem is proved. Assume that $X^*$ or $Z^*$ has the approximation property. If there exists a bounded linear extension operator from $Y^*$ to $Z^*$, then any bounded linear operator $T:X\rightarrow Y$ is nuclear whenever $T$ is nuclear from $X$ to $Z$. The particular case of the theorem with $Z=Y^{**}$ is due to Grothendieck and Oja and Reinov. Numerous applications are presented. For instance, it is shown that a bounded linear operator $T$ from an arbitrary Banach space $X$ to an $\mathcal{L}_\infty$-space $Y$ is nuclear whenever $T$ is nuclear from $X$ to some Banach space $Z$ containing $Y$ as a subspace. AMS 2000 Mathematics subject classification: Primary 46B20; 46B28; 47B10
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source Cambridge Journals Online
subjects approximation property
extension operator
nuclear operators
projective tensor products
title OPERATORS THAT ARE NUCLEAR WHENEVER THEY ARE NUCLEAR FOR A LARGER RANGE SPACE
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