Decomposable quadratic forms in Banach spaces

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. Mathematics Subject Classification (2000). Primary 46B99; Secondary 52A41, 15A63.