On the extensions of homogeneous polynomials
HTML articles powered by AMS MathViewer
- 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
- Richard M. Aron and Paul D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3–24 (English, with French summary). MR 508947
- R. Aron, C. Boyd, and Y. S. Choi, Unique Hahn-Banach theorems for spaces of homogeneous polynomials, J. Aust. Math. Soc. 70 (2001), no. 3, 387–400. MR 1829965, DOI 10.1017/S1446788700002408
- Yun Sung Choi, Kwang Hee Han, and Hyun Gwi Song, Extensions of polynomials on preduals of Lorentz sequence spaces, Glasg. Math. J. 47 (2005), no. 2, 395–403. MR 2203508, DOI 10.1017/S0017089505002624
- A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351–356. MR 947313, DOI 10.1090/S0002-9939-1989-0947313-8
- Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
- C. Hao, A. Kamińska and N. Tomczak-Jaegermann, Orlicz spaces with convexity and concavity constant one, J. Math. Anal. and Appl. 320 (2006), 303–321.
- P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
- Anna Kamińska and Han Ju Lee, On uniqueness of extension of homogeneous polynomials, Houston J. Math. 32 (2006), no. 1, 227–252. MR 2202363
- Anna Kamińska and Han Ju Lee, $M$-ideal properties in Marcinkiewicz spaces, Comment. Math. (Prace Mat.) Tomus specialis in Honorem Juliani Musielak (2004), 123–144. MR 2111760
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
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