Bounded approximation properties via integral and nuclear operators
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- by Åsvald Lima, Vegard Lima and Eve Oja PDF
- Proc. Amer. Math. Soc. 138 (2010), 287-297 Request permission
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 \[ \limsup _\alpha \|TS_\alpha \|_{\mathcal A}\leq \lambda \|T\|_{\mathcal A}.\] 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.References
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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
- MR Author ID: 723061
- 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 287-297
- MSC (2000): Primary 46B28; Secondary 46B20, 47B10, 47L05, 47L20
- DOI: https://doi.org/10.1090/S0002-9939-09-10061-8
- MathSciNet review: 2550194