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

DOI:
https://doi.org/10.1090/S0002-9939-07-08692-3

Published electronically:
March 14, 2007

MathSciNet review:
2302568

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Abstract: We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

**1.**R. Aron and P.D. Berner,*A Hahn-Banach extension theorem for analytic mappings,*Bull. Math. Soc. France.**106**(1978), 3-24.MR**0508947 (80e:46029)****2.**R. Aron, C. Boyd, and Y.S. Choi,*Unique Hahn-Banach theorems for spaces of homogeneous polynomials,*J. Austr. Math. Soc.**70**(2001), 387-400. MR**1829965 (2002g:46076)****3.**Y.S. Choi, K.H. Han, and H.G. Song,*Extensions of polynomials on preduals of Lorentz sequence spaces*, Glasgow Math. J.**47**(2005), 395-403. MR**2203508****4.**A.M. Davie and T.W. Gamelin,*A theorem on polynomial-star approximation,*Proc. Amer. Math. Soc.**106**(1989), 351-356.MR**0947313 (89k:46023)****5.**S. Dineen,*Complex Analysis on Infinite-dimensional Spaces,*Springer-Verlag, London, 1999. MR**1705327 (2001a:46043)****6.**C. Hao, A. Kaminska and N. Tomczak-Jaegermann,*Orlicz spaces with convexity and concavity constant one*, J. Math. Anal. and Appl.**320**(2006), 303-321.**7.**P. Harmand, D. Werner, and W. Werner,*-ideals in Banach Spaces and Banach Algebras*, Lecture Notes in Math.**1547**, Springer-Verlag, 1993. MR**1238713 (94k:46022)****8.**A. Kaminska and H.J. Lee,*On uniqueness of extension of homogeneous polynomials*, Houston J. Math.**32 (1)**(2006), 227-252. MR**2202363****9.**A. Kaminska and H.J. Lee,*-ideal properties in Marcinkiewicz spaces*, Comment. Math., special volume for 75th birthday of Julian Musielak (2004), 123-144.MR**2111760 (2005j:46008)****10.**J. Lindenstrauss and L. Tzafriri,*Classical Banach Spaces I-Sequence Spaces,*Springer-Verlag, 1977. MR**0500056 (58:17766)****11.**J. Lindenstrauss and L. Tzafriri,*Classical Banach Spaces II-Function Spaces,*Springer-Verlag, 1979. MR**0540367 (81c:46001)**

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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:
https://doi.org/10.1090/S0002-9939-07-08692-3

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