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

Authors:
Klaus Floret and Stephan Hunfeld

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

Published electronically:
December 27, 2001

MathSciNet review:
1879966

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -homogeneous polynomials and -linear mappings--scalar-valued and vector-valued. The motivation for these results is the following important special case: the ``uniterated'' Aron-Berner extension : of an -homogeneous polynomial 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 -homogeneous scalar-valued polynomials and ideals of -linear mappings.

**[A1]**R. Alencar,*On reflexivity and basis of*, Proc. Roy. Irish Acad.**85**(1985), 131-138. MR**87i:46101****[A2]**R. Alencar,*An application of Singer's theorem to homogeneous polynomials*, Contemp. Math.**144**(1993), 1-8. MR**94c:53087****[AB]**R. Aron, P. Berner,*A Hahn-Banach extension theorem for analytic mappings*, Bull. Soc. Math. France**106**(1978), 3-24. MR**80e:46029****[CD]**D. Carando, V. Dimant,*Duality in spaces of nuclear and integral polynomials*, J. Math. Anal. Appl.**241**(2000), 107-121. MR**2001c:46089****[CZ]**D. Carando, I. Zalduendo,*A Hahn-Banach theorem for integral polynomials*, Proc. Amer. Math. Soc.**127**(1999), 241-250. MR**99b:46067****[DF]**A. Defant, K. Floret,*Tensor Norms and Operator Ideals*, North Holland Math. Studies 176, 1993. MR**94e:46130****[DG]**A. Davie, T. Gamelin,*A theorem on polynomial-star approximation*, Proc. Amer. Math. Soc.**106**(1989), 351-356. MR**89k:46023****[DT]**S. Dineen, R.M. Timoney,*Complex geodesics on convex domains*, in: Progress Funct. Anal. (eds.: Bierstedt, Bonet, Horvath, Maestre) North Holland Math. Studies**170**(1992), 333-365. MR**92m:46066****[F1]**K. Floret,*Natural norms on symmetric tensor products of normed spaces*, Note di Matematica (Trier-conference 1997)**17**(1997), 153-188. MR**2001g:46038****[F2]**K. Floret,*The metric theory of symmetric tensor products of normed spaces*, in preparation.**[H]**S. Heinrich,*Ultraproducts in Banach space theory*, J. Reine Angew. Math.**313**(1980), 77-104. MR**82b:46013****[JM]**J.A. Jaramillo, L.A. de Moraes,*Duality and reflexivity in spaces of polynomials*, Arch. Math. (Basel)**74**(2000), 282-293. MR**2000k:46063****[K]**K.D. Kürsten,*-Zahlen und Ultraprodukte von Operatoren in Banachräumen*, Doctoral Thesis, Leipzig, 1976.**[KR]**P. Kirwan, R. Ryan,*Extendibility of homogeneous polynomials on Banach spaces*, Proc. AMS**126**(1998), 1023-1029. MR**98f:46042****[LR]**M. Lindström, R. Ryan,*Applications of ultraproducts to infinite dimensional holomorphy*, Math. Scand.**71**(1992), 229-242. MR**94c:46090****[P]**A. Pietsch,*Ideals of multilinear functionals*, Proc. Int. Conf. Operator Alg., etc. (Teubner Texte Math. 62), Leipzig (1984), 185-199. MR**85g:00027**

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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

DOI:
https://doi.org/10.1090/S0002-9939-01-06228-1

Keywords:
Tensor products,
symmetric tensor products,
ideals of polynomials,
ideals of $n$-linear mappings,
ultraproducts

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

Article copyright:
© Copyright 2001
American Mathematical Society