Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces

Floret, Klaus; Hunfeld, Stephan

doi:10.1090/S0002-9939-01-06228-1

Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces
HTML articles powered by AMS MathViewer

by Klaus Floret and Stephan Hunfeld PDF

Proc. Amer. Math. Soc. 130 (2002), 1425-1435 Request permission

Abstract:

Using the theory of full and symmetric tensor norms on normed spaces, a theorem of Kürsten and Heinrich on ultrastability and maximality of normed operator ideals is extended to ideals of $n$-homogeneous polynomials and $n$-linear mappings—scalar-valued and vector-valued. The motivation for these results is the following important special case: the “uniterated” Aron-Berner extension $\overline {q}^{\mathfrak U}$: $E'' \longrightarrow F''$ of an $n$-homogeneous polynomial $q: E\longrightarrow F$ to the bidual remains in certain ideals under preservation of the norm. Moreover, Lotz’s characterization of maximal normed ideals of linear mappings through appropriate tensor norms is proved for ideals of $n$-homogeneous scalar-valued polynomials and ideals of $n$-linear mappings.

References

Raymundo Alencar, On reflexivity and basis for $P(^mE)$, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 131–138. MR 845536
Tsutomu Okada, On a representation of the projective connections of Finsler manifolds, Publ. Math. Debrecen 42 (1993), no. 1-2, 11–27. MR 1208849
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, DOI 10.24033/bsmf.1862
Daniel Carando and Verónica Dimant, Duality in spaces of nuclear and integral polynomials, J. Math. Anal. Appl. 241 (2000), no. 1, 107–121. MR 1738337, DOI 10.1006/jmaa.1999.6626
Daniel Carando and Ignacio Zalduendo, A Hahn-Banach theorem for integral polynomials, Proc. Amer. Math. Soc. 127 (1999), no. 1, 241–250. MR 1458865, DOI 10.1090/S0002-9939-99-04485-8
Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
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 and Richard M. Timoney, Complex geodesics on convex domains, Progress in functional analysis (Peñíscola, 1990) North-Holland Math. Stud., vol. 170, North-Holland, Amsterdam, 1992, pp. 333–365. MR 1150757, DOI 10.1016/S0304-0208(08)70330-X
Klaus Floret, Natural norms on symmetric tensor products of normed spaces, Proceedings of the Second International Workshop on Functional Analysis (Trier, 1997), 1997, pp. 153–188 (1999). MR 1749787
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
J. A. Jaramillo and L. A. Moraes, Duality and reflexivity in spaces of polynomials, Arch. Math. (Basel) 74 (2000), no. 4, 282–293. MR 1742640, DOI 10.1007/s000130050444
Pádraig Kirwan and Raymond A. Ryan, Extendibility of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1023–1029. MR 1415346, DOI 10.1090/S0002-9939-98-04009-X
Mikael Lindström and Raymond A. Ryan, Applications of ultraproducts to infinite-dimensional holomorphy, Math. Scand. 71 (1992), no. 2, 229–242. MR 1212706, DOI 10.7146/math.scand.a-12424
Hellmut Baumgärtel, Gerd Lassner, Albrecht Pietsch, and Armin Uhlmann (eds.), Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 67, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1984. Held in Leipzig, September 25–October 2, 1983. MR 763516