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Title of Journal: Positivity

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Abbravation: Positivity

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Birkhäuser-Verlag

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DOI

10.1016/0920-5632(95)00292-h

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1572-9281

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Odd Degree Polynomials on Real Banach Spaces

Authors: Richard M Aron Petr Hájek
Publish Date: 2006/10/13
Volume: 11, Issue: 1, Pages: 143-153
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Abstract

A classical result of Birch claims that for given k n integers nodd there exists some N = Nk n such that for an arbitrary nhomogeneous polynomial P on Open image in new window there exists a linear subspace Open image in new window of dimension at least k where the restriction of P is identically zero we say that Y is a null space for P Given n 1 odd and arbitrary real separable Banach space X or more generally a space with wseparable dual X we construct an nhomogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ Open image in new window such that every null space containing x has dimension at most k In particular P has no infinite dimensional null space For a given n odd and a cardinal τ we obtain a cardinal N = Nτ n = exp n +1τ such that every nhomogeneous polynomial on a real Banach space X of density N has a null space of density τ Some of the work on this paper was done while the first author was a visitor to the Departamento de Análisis Matemático of the Universidad Complutense de Madrid to which great thanks are given The research of the second author was supported by grants Institutional Research Plan AV0Z10190503 A100190502 GA ČR 201/04/0090


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Other Papers In This Journal:

  1. Order-unit-metric spaces
  2. The Radon–Nikodym Property for Tensor Products of Banach Lattices
  3. Positive bases in spaces of polynomials
  4. Regular orbits and positive directions
  5. Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem
  6. A new application of non-increasing sequences
  7. On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem
  8. A class of $$\ell ^p$$ saturated Banach spaces
  9. On the Structure of the Completion of a Normed Lattice
  10. Domination by ergodic elements in ordered Banach algebras
  11. Pentadiagonal Oscillatory Matrices with Two Spectrum in Common
  12. The essential spectral radius of Volterra operators and an application to Volterra-Hammerstein functional inclusions
  13. An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators
  14. Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality
  15. Statistical approximation by double sequences of positive linear operators on modular spaces
  16. Modules and extremal completely positive maps
  17. Positive solutions of semi-positone nonlinear operator equations in Banach spaces with lattice structure
  18. A characterization of positively decomposable non-linear maps between Banach lattices
  19. Positive solutions for fractional differential systems with nonlocal Riemann–Liouville fractional integral boundary conditions
  20. Coupled coincidence point results on generalized distance in ordered cone metric spaces

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