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

N. Kalton·S. V. Konyagin·L. Vesely
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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 $T\colon X\to X^*$ 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 $L_p(μ)$ 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.

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N. Kalton, S. V. Konyagin, L. Vesely. 2007-08-28. Delta-semidefinite and delta-convex quadratic forms in Banach spaces. https://arxiv.org/abs/math/0605549

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