Archiv der Mathematik

, Volume 110, Issue 6, pp 605–615 | Cite as

Linearization of multipolynomials and applications

  • Geraldo BotelhoEmail author
  • Ewerton R. Torres
  • Thiago Velanga


We prove that every multipolynomial between Banach spaces is the composition of a canonical multipolynomial with a linear operator, and that this correspondence establishes an isometric isomorphism between the spaces of multipolynomials and linear operators. Applications to composition ideals of multipolynomials and to multipolynomials that are of finite rank, approximable, compact, and weakly compact are provided.


Banach spaces Multipolynomials Linearization Projective tensor product Compact and weakly compact operators 

Mathematics Subject Classification

46G25 47H60 47L22 46M05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer, Berlin, 2006.zbMATHGoogle Scholar
  2. 2.
    C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dodrecht, 2006.CrossRefzbMATHGoogle Scholar
  3. 3.
    G. Botelho, D. Pellegrino, and P. Rueda, On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci. 43 (2007), 1139–1155.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Botelho and L. Polac, A polynomial Hutton theorem with applications, J. Math. Anal. Appl. 415 (2014), 294–301.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993.zbMATHGoogle Scholar
  6. 6.
    J. Diestel, H. Jarchow, and A. Pietsch, Operator ideals, In: Handbook of the Geometry of Banach Spaces, Vol. I, 437-496, North-Holland Amsterdam, 2001.CrossRefzbMATHGoogle Scholar
  7. 7.
    K. Floret, Natural norms on symmetric tensor products of normed spaces, Proceedings of the Second International Workshop on Functional Analysis (Trier, 1997), Note Mat. 17 (1997), 153–188 (1999).Google Scholar
  8. 8.
    P. Galindo, D. Garcia, and M. Maestre, Holomorphic mappings of bounded type, J. Math. Anal. Appl. 166 (1992), 236-246.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Mazet, Analytic sets in locally convex spaces, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1984.Google Scholar
  10. 10.
    J. Mujica, Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 324 (1991), 867–887.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Pełczyński, On weakly compact polynomial operators on B-spaces with Dunford-Pettis property, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 371–378.MathSciNetGoogle Scholar
  12. 12.
    A. Pietsch, Operator Ideals, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1980.Google Scholar
  13. 13.
    R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, Thesis, Trinity College Dublin, 1980.Google Scholar
  14. 14.
    R. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002.CrossRefGoogle Scholar
  15. 15.
    T. Velanga, Ideals of polynomials between Banach spaces revisited, to appear in Linear Multilinear Algebra 10.1080/03081087.2017.1394963.Google Scholar
  16. 16.
    T. Velanga, Multilinear mappings versus homogeneous polynomials, arXiv:1706.04703 [math.FA], 2017.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Geraldo Botelho
    • 1
    Email author
  • Ewerton R. Torres
    • 1
  • Thiago Velanga
    • 2
    • 3
  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil
  2. 2.IMECC-UNICAMPUniversidade Estadual de CampinasSão PauloBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de RondôniaPorto VelhoBrazil

Personalised recommendations