Inner and outer inequalities with applications to approximation properties
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Abstract:
Let $X$ be a closed subspace of a Banach space $W$ and let ${\mathcal {F}}$ be the operator ideal of finite-rank operators. If $\alpha$ is a tensor norm, $\mathcal {A}$ is a Banach operator ideal, and $\lambda >0$, then we call the condition â$\|S\|_\alpha \leq \lambda \| S\|_{\mathcal {A}(X,W)} \ \textrm {for\ all}\ S\in \mathcal {F}(X,X)$â an inner inequality and the condition â$\| T\|_\alpha \leq \lambda \| T\|_{\mathcal {A}(Y,W)}$ for all Banach spaces $Y$ and for all $T\in \mathcal {F}(Y,X)$â an outer inequality. We describe cases when outer inequalities are determined by inner inequalities or by some subclasses of Banach spaces. This provides, among others, a unified approach to the study of approximation properties. We present various applications to Grothendieckâs classical approximation properties, to the weak bounded approximation property, and to approximation properties of order $p$.References
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Additional Information
- Eve Oja
- Affiliation: Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia
- Email: eve.oja@ut.ee
- Received by editor(s): July 7, 2008
- Received by editor(s) in revised form: October 14, 2009
- Published electronically: June 8, 2011
- Additional Notes: This research was partially supported by Estonian Science Foundation Grant 7308 and Estonian Targeted Financing Project SF0180039s08.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 5827-5846
- MSC (2010): Primary 46B28; Secondary 46B20, 47B10, 47L05, 47L20
- DOI: https://doi.org/10.1090/S0002-9947-2011-05241-4
- MathSciNet review: 2817411