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

 

On the extensions of homogeneous polynomials


Authors: Anna Kaminska and Pei-Kee Lin
Journal: Proc. Amer. Math. Soc. 135 (2007), 2471-2482
MSC (2000): Primary 46A22, 46A45, 46G25
Published electronically: March 14, 2007
MathSciNet review: 2302568
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Abstract: We investigate the problem of the uniqueness of the extension of $ n$-homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space $ X$ that admits contractive projection of finite rank of at least dimension 2, for every $ n\ge 3$ there exists an $ n$-homogeneous polynomial on $ X$ that has infinitely many extensions to $ X^{**}$. We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice $ E$, for instance when $ E$ satisfies an upper $ p$-estimate with constant one for some $ p>2$, any $ 2$-homogeneous polynomial on $ E$ attaining its norm at $ x\in E$ with a finite rank band projection $ P_x$, has a unique extension to its bidual $ E^{**}$. We apply these results in a class of Orlicz sequence spaces.


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Additional Information

Anna Kaminska
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: kaminska@memphis.edu

Pei-Kee Lin
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: pklin@memphis.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08692-3
PII: S 0002-9939(07)08692-3
Keywords: Hahn-Banach extension, homogeneous polynomials, symmetric sequence spaces
Received by editor(s): December 12, 2005
Received by editor(s) in revised form: March 8, 2006
Published electronically: March 14, 2007
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society