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Inner and outer inequalities with applications to approximation properties


Author: Eve Oja
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
Published electronically: June 8, 2011
MathSciNet review: 2817411
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Abstract | References | Similar Articles | Additional Information

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 `` $ \Vert S\Vert _\alpha\leq\lambda\Vert S\Vert _{\mathcal{A}(X,W)} \textrm{for all} S\in\mathcal{F}(X,X)$'' an inner inequality and the condition `` $ \Vert T\Vert _\alpha\leq\lambda\Vert T\Vert _{\mathcal{A}(Y,W)}$ for all Banach spaces $ Y$ and for all $ T\in\mathcal{F}(Y,X)$'' 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 Grothendieck's classical approximation properties, to the weak bounded approximation property, and to approximation properties of order $ p$.


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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

DOI: https://doi.org/10.1090/S0002-9947-2011-05241-4
Keywords: Banach spaces, finite-rank operators, tensor norms, operator ideals, spaces of operators, approximation properties
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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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