Zeros of polynomials on Banach spaces: The real story

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 7; no. 4; pp. 285 - 295
• Main Authors: ARON, R. M, BOYD, C, RYAN, R. A, ZALDUENDO, I
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01-12-2003; Springer Nature B.V
• Subjects: Algebra; Algebraic geometry; Banach spaces; Exact sciences and technology; Functional analysis; Hilbert space; Mathematical analysis; Mathematics; Polynomials; Sciences and techniques of general use; Several complex variables and analytic spaces; Theory; Dichotomy; Positive definite 2 homogeneous polynomial; Mathematical analysis; Necessary and sufficient condition; Zero of polynomial; Banach space; Compact set; Complex space
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1023/A:1026278115574

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. [PUBLICATION ABSTRACT]