Positivity

, Volume 8, Issue 3, pp 283–296 | Cite as

Weak-Polynomial Convergence on Spaces ℓ p and L p

  • Silvia Lassalle
  • José G. Llavona
Article
  • 36 Downloads

Abstract

This paper is concerned with the study of the set P-1(0), when P varies over all orthogonally additive polynomials on ℓ p and L p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.

polynomials on Banach spaces weak-polynomial topologies zeros of polynomials on ℓp and Lp 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aron R. and Rueda, P. (1997), A problem concerning zero-subspaces of homogeneous polynomials, Linear Topological Spaces and Complex Analysis 3 20–23.Google Scholar
  2. Aron R., Cole, B. and Gamelin, T. (1991), Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math., 415 51–93.Google Scholar
  3. Aron R., Gonzalo, R. and Zagorodnyuk, A. (2000), Zeros of real polynomials. Linear and Multilinear Algebra 48 107–115.Google Scholar
  4. Aron R., Boyd, C., Ryan, R. and Zalduendo, I. Zeros of polynomials on Banach spaces: the real story. To appear in Positivity.Google Scholar
  5. Biström, P., Jaramillo, J.A. and Lindström, M. (1998), Polynomial compactness in Banach spaces, Rocky Mountain Journal of Math. 28 1203–1226.Google Scholar
  6. Bonic R. and Frampton, J. (1966), Smooth functions on Banach Manifolds, J. Math. Mech. 15 877–898.Google Scholar
  7. Carne T.K., Cole, B. and Gamelin, T.W. (1989), A uniform algebra of analytic functions on a Banach space. Trans. Amer. Math. Soc. 314 639–659.Google Scholar
  8. Castillo M.F., García, R. and Gonzalo, R. (1999), Banach spaces in which all multilinear forms are weakly sequentially continuous. Studia Math. 136(2) 121–145.Google Scholar
  9. Davie A.M. and Gamelin, T.W. (1989), A Theorem of polynomial-star approximation. Trans. Amer. Math. Soc. 106 351–358.Google Scholar
  10. Fabián M., Preiss, D. Whitfield, J.H.M. and Zizler, V.E. (1989), Separating polynomials on Banach spaces Quart. J. Math. Oxford (2) 40 409–422.Google Scholar
  11. Garrido M. I., J. A. Jaramillo and J. G. Llavona. Polynomial topologies and uniformities, preprint.Google Scholar
  12. González M., Gutiérrez, J.M. and Llavona, J.G. (1997), Polynomial continuity on ℓ 1, Proc. Amer. Math. Soc. 125(5) 1349–1353.CrossRefGoogle Scholar
  13. Gutiérrez J.M. and Llavona, J.G. (1997), Polynomially continuous operators, Israel J. Math. 102 179–187.Google Scholar
  14. Lindenstrauss, J. and Tzafriri, L. (1977), Classical Banach Spaces II, Springer, Berlin.Google Scholar
  15. Llavona J.G. (1986), Approximation of Continuously Differentiable Functions. North Holland, Amsterdam, Mathematics Studies, 130.Google Scholar
  16. Pelczyński, A. (1957), A property of multilinear operations, Studia Math. 16 173–182.Google Scholar
  17. Royden, H.L. (1988), Real Analysis. 3rd Edition, Collier Macmillan.Google Scholar
  18. Sundaresan, K. (1991), Geometry of spaces of homogeneous polynomials on Banach lattices. Applied Geometry and Discrete Mathematics 571–586, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, Amer. Math. Soc., Prov., RI.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Silvia Lassalle
    • 1
  • José G. Llavona
    • 2
  1. 1.Departamento de Matemática, Facultad de Cs. Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad Complutense de MadridMadridEspaña

Personalised recommendations