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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the extensions of homogeneous polynomials
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by Anna Kamińska and Pei-Kee Lin PDF
Proc. Amer. Math. Soc. 135 (2007), 2471-2482 Request permission

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.
References
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Additional Information
  • Anna Kamińska
  • 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
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2471-2482
  • MSC (2000): Primary 46A22, 46A45, 46G25
  • DOI: https://doi.org/10.1090/S0002-9939-07-08692-3
  • MathSciNet review: 2302568