Decomposable quadratic forms in Banach spaces
Sergei Vladimirovich Konyagin, Libor Veselý · arXiv (Cornell University) · 2006
Abstract. A continuous quadratic form on a real Banach space X is called decomposable if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if X belongs to a certain class of superreflexive Banach spaces, including all Lp(µ) spaces with 2 ≤ p < ∞, then each continuous quadratic form on X is decomposable. On the other hand, on each infinite-dimensional L1(µ) space there exists a continuous quadratic form q that is not delta-convex (i.e., q is not representable as difference of two continuous convex functions); in particular, q is not decomposable. Related results concerning delta-convexity are proved and some open problems are stated.