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

Published online by Cambridge University Press:  09 November 2004

Eve Oja
Affiliation:
Faculty of Mathematics and Computer Science, Tartu University, Liivi 2, EE-50409 Tartu, Estonia (eveoja@math.ut.ee)
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Abstract

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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

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2004