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Existence of polynomials on subspaces without extension

Author: Maite Fernández-Unzueta
Journal: Proc. Amer. Math. Soc. 141 (2013), 2389-2399
MSC (2010): Primary 47H60, 46B07
Published electronically: March 26, 2013
MathSciNet review: 3043020
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Abstract: We prove the existence of a polynomial of degree $ d$ defined on a closed subspace that cannot be extended to the Banach space $ E$ (in particular, the existence of a nonextendible polynomial) in the following cases: (1) $ d\geq 2$ and $ E$ does not have type $ p$ for some $ 1<p<2$; (2) the space $ \ell _k$, $ k\in \mathbb{N}$, $ 2<k\leq d$, is finitely representable in $ E$. In each of these cases we prove, equivalently, the existence of a closed subspace $ F\subset E$ such that the subspace $ \hat {\otimes }^{d}_{s,\pi }{F}$ is not closed in $ \hat {\otimes }^{d}_{s,\pi }{E}$.

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Maite Fernández-Unzueta
Affiliation: Centro de Investigación en Matemáticas (CIMAT), A.P. 402, 36000 Guanajuato, Gto., México

Received by editor(s): February 1, 2011
Received by editor(s) in revised form: October 20, 2011
Published electronically: March 26, 2013
Additional Notes: The author has been partially supported by P48363 CONACyT, México. She would also like to thank the Fields Institute, since part of the research was done while the author was a visiting professor there.
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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