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Extension of polynomials defined on subspaces

Published online by Cambridge University Press:  16 March 2010

MAITE FERNÁNDEZ-UNZUETA
Affiliation:
Centro de Investigación en Matemáticas (CIMAT), A.P. 402, Guanajuato, 36000, Gto., Mexico. e-mail: [email protected]
ÁNGELES PRIETO
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: [email protected]

Abstract

Let k ∈ ℕ and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace FE, there exists a continuous morphism φk, F: (kF) → (kE) satisfying φk, F(P)|F = P, if and only E is isomorphic to a Hilbert space.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Aron, R. M. and Berner, P. D.A Hahn–Banach extension theorem for analytic mappings. Bull. Soc. Math. France 106 (1978), no. 1, 324.CrossRefGoogle Scholar
[2]Aron, R. M., Boyd, C. and Choi, Y. S.Unique Hahn–Banach Theorems for spaces of homogeneous polynomials. J. Aust. Math. Soc. 70 (2001), no. 3, 387400.CrossRefGoogle Scholar
[3]Aron, R. M. and Schottenloher, M.Compact holomorphic mappings on Banach spaces and the approximation theory. J. Funct. Anal. 21 (1976), 730.CrossRefGoogle Scholar
[4]Bennett, G., Dor, L. E., Goodman, V., Johnson, W. B. and Newman, C. M.On uncomplemented subspaces of L p, 1 < p < 2. Israel J. Math. 26 (1977), no. 2, 178187.CrossRefGoogle Scholar
[5]Bourgain, J.A counterexample to a complementation problem. Compositio Math. 43 (1981), no. 1, 133144.Google Scholar
[6]Carando, D.Extendible polynomials on Banach spaces. J. Math. Anal. Math. Appl. 233 (1999), 359372.CrossRefGoogle Scholar
[7]Casazza, P. G. and Nielsen, N. J.The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures. Studia Math. 155 (2003), no. 1, 121.CrossRefGoogle Scholar
[8]Castillo, J. M. F., Defant, A., García, R., Pérez-García, D. and Suárez, J. Local complementation and the extension of bilinear mappings, preprint.Google Scholar
[9]Diestel, J., Jarchow, H. and Tonge, A.Absolutely summing operators Cambridge Studies in Advanced Mathematics, 43. (Cambridge University Press, 1995).CrossRefGoogle Scholar
[10]Davis, W. J., Dean, D. W. and Singer, I.Complemented subspaces and Λ systems in Banach spaces. Israel. J. Math. 6 (1968), 303309.CrossRefGoogle Scholar
[11]Dineen, S. Holomorphically complete locally convex topological vector spaces. Lecture Notes in Math. 332. (Springer, 1973), 77111.Google Scholar
[12]Dineen, S.Complex analysis on infinite dimensional spaces. Springer Monographs in Mathematics (Springer, 1999).CrossRefGoogle Scholar
[13]Fakhoury, H.Sélections linéaires associées au théorème de Hahn–Banach. J. Funct. Anal. 11 (1972), 436452.CrossRefGoogle Scholar
[14]Figiel, T., Lindenstrauss, J. and Milman, V. D.The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), no. 1-2, 5394.CrossRefGoogle Scholar
[15]Galindo, P., García, D., Maestre, M. and Mujica, J.Extension of multilinear mappings on Banach spaces. Studia Math. 108 (1994), no. 1, 5576.CrossRefGoogle Scholar
[16]Garling, D. J. H. and Gordon, Y.Relations between some constants associated with finite dimensional Banach spaces. Israel J. Math. 9 (1971), 346361.CrossRefGoogle Scholar
[17]Jarchow, H., Palazuelos, C., Pérez-García, D. and Villanueva, I.Hahn–Banach extension of multilinear forms and summability. J. Math. Anal. Appl. 336 (2007), no. 2, 11611177.CrossRefGoogle Scholar
[18]Kirwan, P. and Ryan, R.Extendibility of homogeneous polynomials on Banach spaces. Proc. Amer. Math. Soc. 126 (4) (1998), 10231029.CrossRefGoogle Scholar
[19]Lindenstrauss, J. and Tzafriri, L.On the complemented subspaces problem. Israel. J. Math. 9 (1971), 263269.CrossRefGoogle Scholar
[20]Lindenstrauss, J. and Tzafriri, L.Classical Banach spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92 (Springer-Verlag, 1977).Google Scholar
[21]Maurey, B.Un Théorème de prolongement. C. R. Acad. Sci. Paris Sér. A 279 (1974), 329332.Google Scholar
[22]Mujica, J.Complex analysis in Banach spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions. North-Holland Mathematics Studies, 120. (North-Holland Publishing Co., 1986).Google Scholar
[23]Pełczyński, A.On weakly compact polynomial operators on B-spaces with Dunford-Pettis property. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 11 (1963), 371378.Google Scholar
[24]Pietsch, A.History of Banach spaces and linear operators (Birkhäuser Boston, Inc., 2007).Google Scholar
[25]Rosenthal, H. P.On the subspaces of L p (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8 (1970), 273303.CrossRefGoogle Scholar
[26]Zalduendo, I.Extending polynomials on Banach spaces–a survey. Rev. Un. Mat. Argentina 46 (2005), no. 2, 4572.Google Scholar