Extension of vector-valued integral polynomials. (English) Zbl 1082.46034

Carando, Daniel; Lassalle, Silvia

J. Math. Anal. Appl. 307, No. 1, 77-85 (2005).

P. Kirwan and R. Ryan [Proc. Am. Math. Soc. 126, 1023–1029 (1998; Zbl 0890.46032)] were perhaps the first to discuss extendibility of polynomials on Banach spaces. An \(n\)-homogeneous polynomial \(P:E \to F\) between Banach spaces is said to be {extendible} if for any larger Banach space \(F \supset E,\) there is \(\widetilde{P} \in {\mathcal P}(^nF,X)\) that extends \(P.\) The space of extendible \(n\)-homogeneous polynomials is denoted \(\mathcal{P}_e(^nE,X).\)
The basic question studied here is whether \(\widetilde{P}\) has a particular property if \(P\) does, and the particular property in question is some kind of integral condition. For instance, Pietsch-integral polynomials are always extendible to polynomials that are Pietsch-integral on the larger space. On the other hand, although Grothendieck-integral polynomials are extendible, these extensions are not in general Grothendieck-integral. Also, provided that \(E\) does not contain a copy of \(\ell_1,\) the canonical extension \(\widetilde{P}\) of any (Grothendieck or Pietsch)-integral polynomial with representing measure \(G\) is given by \(\widetilde{P}(z) = \int_{B_E^\prime} z(\gamma)^n dG(\gamma).\) In fact, using results by the first author and I. Zalduendo [Proc. Am. Math. Soc. 127, 241–250 (1999; Zbl 0908.46031)], the authors obtain an integral representation for \(\widetilde{P}\) that involves the measures that represent \(P.\)

Reviewer: Richard M. Aron (Kent/Ohio)

MSC:

46G25	(Spaces of) multilinear mappings, polynomials
46B28	Spaces of operators; tensor products; approximation properties
46G20	Infinite-dimensional holomorphy

Keywords:

integral polynomials; extendibility

Citations:

Zbl 0890.46032; Zbl 0908.46031

References:

[1]	Alencar, R., Multilinear mappings of nuclear and integral type, Proc. Amer. Math. Soc., 94, 33-38 (1985) · Zbl 0537.47011
[2]	Alencar, R., On reflexivity and basis for \(P({}^mE)\), Proc. Roy. Irish Acad. Sect. A, 85, 131-138 (1985) · Zbl 0594.46043
[3]	Arens, R., The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2, 839-848 (1951) · Zbl 0044.32601
[4]	Aron, R.; Berner, P., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France, 106, 3-24 (1978) · Zbl 0378.46043
[5]	Carando, D., Extendible polynomials on Banach spaces, J. Math. Anal. Appl., 233, 359-372 (1999) · Zbl 0942.46029
[6]	Carando, D.; Dimant, V., Duality of spaces of nuclear and integral polynomials, J. Math. Anal. Appl., 241, 107-121 (2000) · Zbl 0993.46007
[7]	Carando, D.; Lassalle, S., \(E^\prime\) and its relation with vector-valued functions on \(E\), Ark. Mat., 42, 283-300 (2004) · Zbl 1058.46024
[8]	Carando, D.; Zalduendo, I., A Hahn-Banach theorem for integral polynomials, Proc. Amer. Math. Soc., 127, 241-250 (1999) · Zbl 0908.46031
[9]	Castillo, J.; García, R.; Jaramillo, J. A., Extension of bilinear forms on Banach spaces, Proc. Amer. Math. Soc., 129, 3647-3656 (2001) · Zbl 0988.46010
[10]	Cilia, R.; D’Anna, M.; Gutiérrez, J. M., Polynomial characterization of \(L_\infty \)-spaces, J. Math. Anal. Appl., 275, 900-912 (2002) · Zbl 1031.46051
[11]	Cilia, R.; Gutiérrez, J. M., Nuclear and integral polynomials, J. Austral. Math. Soc., 76, 269-280 (2004) · Zbl 1073.46037
[12]	Davie, A. M.; Gamelin, T. W., A theorem of polynomial-star approximation, Trans. Amer. Math. Soc., 106, 351-358 (1989) · Zbl 0683.46037
[13]	Defant, A.; Floret, K., Tensor Norms and Operator Ideals, Math. Studies, vol. 176 (1993), North-Holland: North-Holland Amsterdam · Zbl 0774.46018
[14]	Dineen, S., Complex Analysis in Locally Convex Spaces, Math. Studies, vol. 57 (1981), North-Holland: North-Holland Amsterdam · Zbl 0484.46044
[15]	Diestel, J.; Uhl, J. J., Vector Measures, Math. Surveys Monogr., vol. 15 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0369.46039
[16]	Floret, K., Natural norms on symmetric tensor products of normed spaces, Note Mat., 17, 153-188 (1997) · Zbl 0961.46013
[17]	Galindo, P.; García, D.; Maestre, M.; Mujica, J., Extension of multilinear mappings on Banach spaces, Studia Math., 108, 55-76 (1994) · Zbl 0852.46004
[18]	González, M.; Gutiérrez, J. M., Extension and lifting of polynomials, Arch. Math., 81, 431-438 (2003) · Zbl 1047.46035
[19]	Haydon, R., Some more characterizations of Banach spaces containing \(\ell_1\), Math. Proc. Cambridge Philos. Soc., 80, 269-276 (1976) · Zbl 0335.46011
[20]	Kirwan, P.; Ryan, R., Extendibility of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc., 126, 1023-1029 (1998) · Zbl 0890.46032
[21]	Mazet, P., A Hahn-Banach theorem for quadratic forms, Proc. Roy. Irish Acad. Sect. A, 100, 177-182 (2000) · Zbl 0995.46006
[22]	Mujica, J., Complex Analysis in Banach Spaces, Math. Studies, vol. 120 (1986), North-Holland: North-Holland Amsterdam · Zbl 0586.46040
[23]	R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, PhD thesis, University College, Dublin, 1980; R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, PhD thesis, University College, Dublin, 1980
[24]	Ryan, R., Introduction to Tensor Products of Banach Spaces, Springer Monogr. Math. (2002), Springer-Verlag: Springer-Verlag London · Zbl 1090.46001
[25]	Villanueva, I., Integral mappings between Banach spaces, J. Math. Anal. Appl., 279, 56-70 (2003) · Zbl 1030.47503
[26]	Zalduendo, I., Extending Polynomials (1998), Publ. Dep. Análisis Mat.: Publ. Dep. Análisis Mat. Universidad Complutense de Madrid