This is an announcement for the paper "Decomposable quadratic forms in Banach spaces" by S.V. Konyagin and L. Vesely.

Abstract: A continuous quadratic form on a real Banach space $X$ is called {\em decomposable} if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if $X$ belongs to a certain class of superreflexive Banach spaces, including all $L_p(\mu)$ spaces with $2\le p<\infty$, then each continuous quadratic form on $X$ is decomposable. On the other hand, on each infinite-dimensional $L_1(\mu)$ space there exists a continuous quadratic form $q$ that is not delta-convex (i.e., $q$ is not representable as difference of two continuous convex functions); in particular, $q$ is not decomposable. Related results concerning delta-convexity are proved and some open problems are stated.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46B99 (Primary) 52A41, 15A63 (Secondary)

Remarks: 11 pages

The source file(s), KonyaginVesely.tex: 32898 bytes, birkmult.cls: 53923 bytes, is(are) stored in gzipped form as 0605549.tar.gz with size 26kb. The corresponding postcript file has gzipped size 56kb.

Submitted from: Libor.Vesely@mat.unimi.it

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