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

2006/05/19 by N. Kalton, Kalton, N., S. V. Konyagin +3
Mathematics · #15A63 (Secondary) #46B99 (Primary) #52A41 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:15A63 #msc:46B99 #msc:52A41

paper · pdf · doi:10.48550/arxiv.math/0605549

19 pages

arxiv created 2007/08/28 · arxiv updated 2009/12/01

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\colon 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_p(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 problems are stated.

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