Extendibility of homogeneous polynomials on Banach spaces
Authors:
Pádraig Kirwan and Raymond A. Ryan
Journal:
Proc. Amer. Math. Soc. 126 (1998), 10231029
MSC (1991):
Primary 46G20; Secondary 46B28
MathSciNet review:
1415346
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible homogeneous polynomials on and we characterize the extendible 2homogeneous polynomials on when is a Hilbert space, an space or an space.
 1.
Richard
M. Aron, Extension and lifting theorems for analytic mappings,
Functional analysis: surveys and recent results, II (Proc. Second Conf.
Functional Anal., Univ. Paderborn, Paderborn, 1979) NorthHolland Math.
Stud., vol. 38, NorthHolland, AmsterdamNew York, 1980,
pp. 257–267. MR 565409
(81i:46006)
 2.
Richard
M. Aron and Paul
D. Berner, A HahnBanach extension theorem for analytic
mappings, Bull. Soc. Math. France 106 (1978),
no. 1, 3–24 (English, with French summary). MR 508947
(80e:46029)
 3.
Andreas
Defant and Klaus
Floret, Tensor norms and operator ideals, NorthHolland
Mathematics Studies, vol. 176, NorthHolland Publishing Co.,
Amsterdam, 1993. MR 1209438
(94e:46130)
 4.
A.
M. Davie and T.
W. Gamelin, A theorem on polynomialstar
approximation, Proc. Amer. Math. Soc.
106 (1989), no. 2,
351–356. MR
947313 (89k:46023), http://dx.doi.org/10.1090/S00029939198909473138
 5.
Joe
Diestel, Hans
Jarchow, and Andrew
Tonge, Absolutely summing operators, Cambridge Studies in
Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge,
1995. MR
1342297 (96i:46001)
 6.
Seán
Dineen and Richard
M. Timoney, Complex geodesics on convex domains, Progress in
functional analysis (Peñíscola, 1990) NorthHolland Math.
Stud., vol. 170, NorthHolland, Amsterdam, 1992,
pp. 333–365. MR 1150757
(92m:46066), http://dx.doi.org/10.1016/S03040208(08)70330X
 7.
Pablo
Galindo, Domingo
García, Manuel
Maestre, and Jorge
Mujica, Extension of multilinear mappings on Banach spaces,
Studia Math. 108 (1994), no. 1, 55–76. MR 1259024
(95f:46072)
 8.
Mikael
Lindström and Raymond
A. Ryan, Applications of ultraproducts to infinitedimensional
holomorphy, Math. Scand. 71 (1992), no. 2,
229–242. MR 1212706
(94c:46090)
 9.
P. Mazet, A HahnBanach theorem for quadratic forms, preprint.
 10.
Luiza
A. Moraes, A HahnBanach extension theorem for some holomorphic
functions, Complex analysis, functional analysis and approximation
theory (Campinas, 1984) NorthHolland Math. Stud., vol. 125,
NorthHolland, Amsterdam, 1986, pp. 205–220. MR 893417
(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.
Ignacio
Zalduendo, A canonical extension for analytic
functions on Banach spaces, Trans. Amer. Math.
Soc. 320 (1990), no. 2, 747–763. MR 1001952
(90k:46108), http://dx.doi.org/10.1090/S0002994719901001952X
 1.
 R. Aron, Extension and lifting theorems for analytic mappings, Functional Analysis: Surveys and Recent Results II, Math. Stud. 38, NorthHolland, 1980, 257267. MR 81i:46006
 2.
 R. Aron and P. Berner, A HahnBanach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 324. MR 80e:46029
 3.
 A. Defant and K. Floret, Tensor Norms and Operator Ideals, NorthHolland Math. Studies 176, 1993. MR 94e:46130
 4.
 A. M. Davie and T. W. Gamelin, A theorem on polynomialstar approximation, Proc. Amer. Math. Soc. 106 (1989), 351356. 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, NorthHolland, 1992, 333365. MR 92m:46066
 7.
 P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), 5576. MR 95f:46072
 8.
 M. Lindström and R. A. Ryan, Applications of ultraproducts to infinite dimensional holomorphy, Math. Scand. 71 (1992), 229242. MR 94c:46090
 9.
 P. Mazet, A HahnBanach theorem for quadratic forms, preprint.
 10.
 L. Moraes, A HahnBanach extension theorem for some holomorphic functions, Complex Analysis, Functional Analysis and Approximation Theory (ed. J. Mujica), Math. Studies 125, NorthHolland, 1986, 205220. 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), 747763. MR 90k:46108
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
46G20,
46B28
Retrieve articles in all journals
with MSC (1991):
46G20,
46B28
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
Email:
pkirwan@staffmail.wit.ie
Raymond A. Ryan
Affiliation:
Department of Mathematics, University College, Galway, Ireland
Email:
ray.ryan@ucg.ie
DOI:
http://dx.doi.org/10.1090/S000299399804009X
PII:
S 00029939(98)04009X
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
