  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 $T\colon X\to X^*$ 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.
