Abstract
Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.
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Aron, R., Boyd, C., Ryan, R. et al. Zeros of Polynomials on Banach Spaces: The Real Story. Positivity 7, 285–295 (2003). https://doi.org/10.1023/A:1026278115574
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DOI: https://doi.org/10.1023/A:1026278115574