Positivity 12 (2008), 221–240
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020221-20, published online January 11, 2008
Delta-semidefinite and Delta-convex Quadratic
Forms in Banach Spaces
Nigel Kalton, Sergei V. Konyagin and Libor Vesel´y
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 → 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
(µ) space (1 ≤ p ≤∞) is: (a) delta-semidefinite
iff p ≥ 2; (b) delta-convex iff p>1. Some other related results concerning
delta-convexity are proved and some open probms are stated.
Mathematics Subject Classiﬁcation (2000). Primary 46B99, Secondary 52A41,
Keywords. Banach space, continuous quadratic form, positively semidefinite
quadratic form, delta-semidefinite quadratic form, delta-convex function,
Let X be a real Banach space. Recall that a function q : X → R is a continu-
ous quadratic form (more precise would be “continuous purely quadratic form”)
if there exists a continuous bilinear form b : X × X → R such that q(x)=b(x, x)
for each x ∈ X.
In the present paper, we are interested mainly in the following two isomorphic
properties of X.
(D) Each continuous quadratic form on X is delta-semidefinite, i.e., it can be
represented as a difference of two nonnegative continuous quadratic forms.
(dc) Each continuous quadratic form on X is delta-convex, i.e., it can be repre-
sented as a difference of two continuous convex functions.
The ﬁrst author was supported by NSF grant DMS-0555670. The second author was supported by
the Russian Foundation for Basic Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1.
The third author was supported in part by the Ministero dell’Universit`a e della Ricerca of Italy.