Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces
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- by Klaus Floret and Stephan Hunfeld PDF
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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
Additional Information
- Klaus Floret
- Affiliation: Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany
- Email: floret@mathematik.uni-oldenburg.de
- Stephan Hunfeld
- Affiliation: Werstener Dorfstrasse 209, D-40591 Düsseldorf, Germany
- Received by editor(s): February 9, 1999
- Received by editor(s) in revised form: November 20, 2000
- Published electronically: December 27, 2001
- Communicated by: Dale Alspach
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1425-1435
- MSC (2000): Primary 46B08; Secondary 46B28, 46G25
- DOI: https://doi.org/10.1090/S0002-9939-01-06228-1
- MathSciNet review: 1879966