Zeros of polynomials on Banach spaces: The real story
Let \(E\) be a real Banach space. A 2-homogeneous polynomial \(P\) on \(E\) is positive definite if \(P(x) \geq 0\) for every \(x\) in \(E\) and \(P(x)=0\) implies that \(x=0\). If \(E\) is separable, then \(E\) admits a positive definite 2-homogeneous polynomial. The authors prove three equivalent conditions to the fact that a Banach space admits a positive definite 2-homogeneous polynomial, and then they move on to prove some dichotomy results. It is shown that every 2-homogeneous polynomial defined on a Banach space which does not admit a positive definite 2-homogeneous polynomial vanishes on an infinite-dimensional subspace. The authors conjecture that this subspace is always non-separable and prove this whenever \(E\) is of type 2 or does not admit a positive definite 4-homogeneous polynomial. In this last case, they also show that every 3-homogeneous polynomial is identically zero on a non-separable subspace. The same techniques can be used to show that a polynomial (not necessarily homogeneous) with \(P(0)=0\), defined on a space which does not admit a positive definite homogeneous plynomial of any degree, vanishes on a non-separable subspace. Then a few special cases (like \(E=C(K)\)) are treated and more precise results are obtained, including two dichotomy results concerning nuclear and absolutely \((1,2)\)-summing 2-homogeneous polynomials.
scientific article; zbMATH DE number 2021993