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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounded approximation properties via integral and nuclear operators


Authors: Åsvald Lima, Vegard Lima and Eve Oja
Journal: Proc. Amer. Math. Soc. 138 (2010), 287-297
MSC (2000): Primary 46B28; Secondary 46B20, 47B10, 47L05, 47L20
Published electronically: August 25, 2009
MathSciNet review: 2550194
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Abstract: Let $ X$ be a Banach space and let $ \mathcal A$ be a Banach operator ideal. We say that $ X$ has the $ \lambda$-bounded approximation property for $ \mathcal A$ ($ \lambda$-BAP for $ \mathcal A$) if for every Banach space $ Y$ and every operator $ T\in \mathcal A(X,Y)$, there exists a net $ (S_\alpha)$ of finite rank operators on $ X$ such that $ S_\alpha\to I_X$ uniformly on compact subsets of $ X$ and

$\displaystyle \limsup_\alpha\Vert TS_\alpha\Vert _{\mathcal A}\leq\lambda\Vert T\Vert _{\mathcal A}.$

We prove that the (classical) $ \lambda$-BAP is precisely the $ \lambda$-BAP for the ideal $ \mathcal I$ of integral operators, or equivalently, for the ideal $ {\mathcal{S{\kern -0.15em}I}}$ of strictly integral operators. We also prove that the weak $ \lambda$-BAP is precisely the $ \lambda$-BAP for the ideal $ \mathcal N$ of nuclear operators.


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Additional Information

Åsvald Lima
Affiliation: Department of Mathematics, University of Agder, Serviceboks 422, 4604 Kristiansand, Norway
Email: Asvald.Lima@uia.no

Vegard Lima
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Address at time of publication: Aalesund University College, Service Box 17, N-6025 Ålesund, Norway
Email: lima@math.missouri.edu, Vegard.Lima@gmail.com

Eve Oja
Affiliation: Faculty of Mathematics and Computer Science, University of Tartu, J. Liivi 2, EE-50409 Tartu, Estonia
Email: eve.oja@ut.ee

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10061-8
PII: S 0002-9939(09)10061-8
Keywords: Banach spaces, Banach operator ideals, bounded approximation properties
Received by editor(s): April 17, 2009
Received by editor(s) in revised form: May 29, 2009
Published electronically: August 25, 2009
Additional Notes: The research of the third author was partially supported by Estonian Science Foundation Grant No. 7308
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.