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Extendibility of homogeneous polynomials
on Banach spaces

Authors: Pádraig Kirwan and Raymond A. Ryan
Journal: Proc. Amer. Math. Soc. 126 (1998), 1023-1029
MSC (1991): Primary 46G20; Secondary 46B28
MathSciNet review: 1415346
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Abstract: We study the $n$-homogeneous polynomials on a Banach space $X$ that can be extended to any space containing $X$. We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible $n$-homogeneous polynomials on $X$ and we characterize the extendible 2-homogeneous polynomials on $X$ when $X$ is a Hilbert space, an $\mathcal L_1$-space or an $\mathcal L_\infty$-space.

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  • 1. R. Aron, Extension and lifting theorems for analytic mappings, Functional Analysis: Surveys and Recent Results II, Math. Stud. 38, North-Holland, 1980, 257-267. MR 81i:46006
  • 2. R. Aron and P. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3-24. MR 80e:46029
  • 3. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Studies 176, 1993. MR 94e:46130
  • 4. A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351-356. MR 89k:46023
  • 5. J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press, 1995. MR 96i:46001
  • 6. S. Dineen and R. Timoney, Complex geodesics on convex domains, Progress in Functional Analysis (ed. K. Bierstedt, J. Bonet, J. Horvath and M. Maestre), Math. Studies 170, North-Holland, 1992, 333-365. MR 92m:46066
  • 7. P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), 55-76. MR 95f:46072
  • 8. M. Lindström and R. A. Ryan, Applications of ultraproducts to infinite dimensional holomorphy, Math. Scand. 71 (1992), 229-242. MR 94c:46090
  • 9. P. Mazet, A Hahn-Banach theorem for quadratic forms, preprint.
  • 10. L. Moraes, A Hahn-Banach extension theorem for some holomorphic functions, Complex Analysis, Functional Analysis and Approximation Theory (ed. J. Mujica), Math. Studies 125, North-Holland, 1986, 205-220. MR 88f:46094
  • 11. R. A. Ryan, Applications of Topological Tensor Products to Infinite Dimensional Holomorphy, Ph.D. Thesis, Trinity College, Dublin, 1980.
  • 12. R. A. Ryan and J. B. Turret, Products of linear functionals, Preprint, 1995.
  • 13. I. Zalduendo, A canonical extensions for analytic functions on Banach spaces, Trans. Amer. Math. Soc. 320 (1990), 747-763. MR 90k:46108

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Additional Information

Pádraig Kirwan
Affiliation: Department of Mathematics, University College, Galway, Ireland
Address at time of publication: Department of Physical and Quantitative Sciences, Waterford Institute of Technology, Waterford, Ireland

Raymond A. Ryan
Affiliation: Department of Mathematics, University College, Galway, Ireland

Keywords: Homogeneous polynomial, extendibility
Received by editor(s): May 17, 1996
Received by editor(s) in revised form: July 10, 1996
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society

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