Delta-semidefinite and delta-convex quadratic forms in Banach spaces

Abstract

A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T:X\to X^{\ast }$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(\mu )$ space ($1\le p\le \infty$) is: (a) delta-semidefinite iff $p\ge 2$; (b) delta-convex iff $p>1$. Some other related results concerning delta-convexity are proved and some open problems are stated.

19 pages

Cited by in corpus (1)

• On compositions of d.c. functions and mappings