Abstract

This paper is concerned with the study of MM-structures in spaces of polynomials. More precisely, we discuss for EE and FF Banach spaces, whether the class of weakly continuous on bounded sets nn-homogeneous polynomials, Pw(nE,F)\mathcal P_w(^n E, F), is an MM-ideal in the space of continuous nn-homogeneous polynomials P(nE,F)\mathcal P(^n E, F). We show that there is some hope for this to happen only for a finite range of values of nn. We establish sufficient conditions under which the problem has positive and negative answers and use the obtained results to study the particular cases when E=ℓpE=\ell_p and F=ℓqF=\ell_q or FF is a Lorentz sequence space d(w,q)d(w,q). We extend to our setting the notion of property (M)(M) introduced by Kalton which allows us to lift MM-structures from the linear to the vector-valued polynomial context. Also, when Pw(nE,F)\mathcal P_w(^n E, F) is an MM-ideal in P(nE,F)\mathcal P(^n E, F) we prove a Bishop-Phelps type result for vector-valued polynomials and relate norm-attaining polynomials with farthest points and remotal sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Verónica Dimant

Dep. de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires – Argentina and: CONICET

vero@udesa.edu.ar

Silvia Lassalle

Dep. de Matemática - Pab I, Fac. de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
and: CONICET

slassall@dm.uba.ar

Keywords

  • M-ideals
  • homogeneous polynomials
  • weakly continuous polynomials on bounded sets

Mathematics Subject Classification

47H60,46B04,47L22,46B20

References

  1. M. D. Acosta: Denseness of norm attaining mappings, RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 100(1–2) (2006) 9–30.Google Scholar
  2. M. D. Acosta, F. J. Aguirre, R. Payá: There is no bilinear Bishop-Phelps theorem, Isr. J. Math. 93 (1996) 221–227.https://doi.org/10.1007/BF02761104Google Scholar
  3. E. Alfsen, E. Effros: Structure in real Banach spaces. I, II. Ann. of Math. (2)96 (1972) 98–128; Ann. of Math. (2)96 (1972) 129–173.https://doi.org/10.2307/1970895https://doi.org/10.1176/ajp.129.1.96Google Scholar
  4. R. Alencar, K. Floret: Weak-strong continuity of multilinear mappings and the Pelczynski-Pitt theorem, J. Math. Anal. Appl. 206(2) (1997) 532–546.https://doi.org/10.1006/jmaa.1997.5253Google Scholar
  5. R. Aron, P. Berner: A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. Fr. 106 (1978) 3–24.https://doi.org/10.24033/bsmf.1862Google Scholar
  6. R. Aron, V. Dimant: Sets of weak sequential continuity for polynomials, Indag. Math., New Ser. 13(3) (2002) 287–299.https://doi.org/10.1016/S0019-3577(02)80012-XGoogle Scholar
  7. R. Aron, D. García, M. Maestre: On norm attaining polynomials, Publ. Res. Inst. Math. Sci. 39(1) (2003) 165–172.https://doi.org/10.2977/prims/1145476151Google Scholar
  8. R. Aron, C. Hervés, M. Valdivia: Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983) 189–204.https://doi.org/10.1016/0022-1236(83)90081-2Google Scholar
  9. R. Aron, J. B. Prolla: Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980) 195–216.https://doi.org/10.1515/crll.1980.313.195Google Scholar
  10. R. M. Aron, M. Schottenloher: Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21(1) (1976) 7–30.https://doi.org/10.1016/0022-1236(76)90026-4Google Scholar
  11. E. Asplund: Farthest points in reflexive locally uniformly rotund Banach spaces, Isr. J. Math. 4 (1966) 213–216.https://doi.org/10.1007/BF02771633Google Scholar
  12. P. Bandyopadhyay, B.-L. Lin, T. S. S. R. K. Rao: Ball remotal subspaces of Banach spaces, Colloq. Math. 114(1) (2009) 119–133.https://doi.org/10.4064/cm114-1-10Google Scholar
  13. C. Boyd, R. A. Ryan: Bounded weak continuity of homogeneous polynomials at the origin, Arch. Math. (Basel) 71(3) (1998) 211–218.https://doi.org/10.1007/s000130050254Google Scholar
  14. D. Carando, S. Lassalle: E ′ and its relation with vector-valued functions on E, Ark. Mat. 42(2) (2004) 283–300.https://doi.org/10.1007/BF02385480Google Scholar
  15. Y. S. Choi, S. G. Kim: Norm or numerical radius attaining multilinear mappings and polynomials, J. Lond. Math. Soc., II. Ser. 54(1) (1996) 135–147.https://doi.org/10.1112/jlms/54.1.135Google Scholar
  16. V. Dimant: M -ideals of homogeneous polynomials, Stud. Math. 202 (2011) 81–104.https://doi.org/10.4064/sm202-1-5Google Scholar
  17. V. Dimant, R. Gonzalo: Block diagonal polynomials, Trans. Amer. Math. Soc. 353(2) (2001) 733–747.https://doi.org/10.1090/S0002-9947-00-02735-5Google Scholar
  18. S. Dineen: Complex Analysis on Infinite-Dimensional Spaces, Springer Monographs in Mathematics, Springer, London (1999).https://doi.org/10.1007/978-1-4471-0869-6Google Scholar
  19. N. Dunford, J. T. Schwartz: Linear Operators. Part I: General Theory, Wiley, New York (1958).Google Scholar
  20. M. Edelstein: Farthest points of sets in uniformly convex Banach spaces, Isr. J. Math. 4 (1966) 171–176.https://doi.org/10.1007/BF02760075Google Scholar
  21. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics / Ouvrages de Mathématiques de la SMC 8, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3480-5Google Scholar
  22. M. González, J. M. Gutiérrez: The polynomial property (V), Arch. Math. 75(4) (2000) 299–306.https://doi.org/10.1007/s000130050507Google Scholar
  23. R. Gonzalo, J. A. Jaramillo: Compact polynomials between Banach spaces, Proc. R. Ir. Acad., Sect. A 95(2) (1995) 213–226.Google Scholar
  24. P. Harmand, D. Werner, W. Werner: M -Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer, Berlin (1993).https://doi.org/10.1007/BFb0084355Google Scholar
  25. J. Hennefeld: M -ideals, HB-subspaces, and compact operators, Indiana Univ. Math. J. 28(6) (1979) 927–934.https://doi.org/10.1512/iumj.1979.28.28065Google Scholar
  26. N. J. Kalton: M -ideals of compact operators, Ill. J. Math. 37(1) (1993) 147–169.https://doi.org/10.1215/ijm/1255987254Google Scholar
  27. N. J. Kalton, D. Werner: Property (M), M -ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995) 137–178.https://doi.org/10.1515/crll.1995.461.137Google Scholar
  28. A. Lima: M -ideals of compact operators in classical Banach spaces, Math. Scand. 44(1) (1979) 207–217.https://doi.org/10.7146/math.scand.a-11804Google Scholar
  29. J. Mujica: Complex Analysis in Banach Spaces, North-Holland Mathematics Studies 120, North-Holland, Amsterdam (1986).Google Scholar
  30. E. Oja: On M -ideals of compact operators and Lorentz sequence spaces, Proc. Est. Acad. Sci., Phys. Math. 40(1) (1991) 31–36.https://doi.org/10.3176/phys.math.1991.1.05Google Scholar
  31. E. Oja, D. Werner: Remarks on M -ideals of compact operators on X ⊕p X, Math. Nachr. 152 (1991) 101–111.https://doi.org/10.1002/mana.19911520110Google Scholar
  32. D. Werner: M -ideals and the “basic inequality”, J. Approximation Theory 76(1) (1994) 21–30.https://doi.org/10.1006/jath.1994.1002Google Scholar
  33. P. Wojtaszczyk: Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge University Press, Cambridge (1991).https://doi.org/10.1017/CBO9780511608735Google Scholar

Suggested citation

V. Dimant, S. Lassalle. “M-Structures in Vector-Valued Polynomial Spaces.” Journal of Convex Analysis 19 (2012), No. 3, 685–711. https://doi.org/10.68381/jca19037

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.