an:05292839 Zbl 1153.46006 Kalton, Nigel; Konyagin, Sergei V.; Veselý, Libor Delta-semidefinite and delta-convex quadratic forms in Banach spaces EN Positivity 12, No. 2, 221-240 (2008). 1385-1292 1572-9281 2008
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46B09 46B03 52A41 15A63 Banach space; continuous quadratic form; positively semi-definite quadratic form; delta-semidefinite quadratic form; delta-convex function; Walsh-Paley martingale; UMD-space In what follows, all the mappings will be supposed continuous, so we shall omit this term. A~quadratic form on a real Banach space $$X$$ is a mapping $$q:X\to \mathbb R$$ such that $$q(x)=b(x,x),\;x\in X,\,$$ for some bilinear form $$b:X\times X \to \mathbb R$$. For given $$q$$, there exists a unique symmetric bilinear form $$\tilde b$$ such that $$q(x)=\tilde b(x,x),\;x\in X,\,$$ and for every symmetric bilinear form $$b$$ on $$X$$, there exists a unique symmetric linear operator $$T:X \to X^*$$ such that $$b(x,y)=\langle Tx,y\rangle,\;x,y\in X$$ (symmetric means that $$T=T^*$$ on $$X$$). One says that the Banach space $$X$$ has the property (D) if every quadratic form on $$X$$ is semi-definite, i.e., it can be written as the difference of two positive quadratic forms; $$X$$ has the property (dc) if every quadratic form on $$X$$ is delta-convex, i.e., it can be written as the difference of two convex functions, and $$X$$ has the property (Cdc) if every $$C^{1,1}$$-function on $$X$$ is delta-convex. The paper is concerned with characterizations of Banach spaces having these properties. Obviously, (D) $$\Rightarrow$$ (dc), but the converse is not true, as is shown by an example. The authors show that a quadratic form $$q$$ is semi-definite iff the symmetric operator $$T$$ corresponding to the symmetric bilinear form generating $$q$$ is factorizable through a Hilbert space. The space $$\ell_p$$ fails to have property (dc) for every $$p,\;1\leq p<2$$. The characterization, given in Section~2, of Banach spaces having property (dc) is more complicated and involves UMD-operators and UMD-spaces in the sense of Burkholder. In particular, every UMD-space has property (dc). The third (and last) section of the paper is concerned with the related property (Cdc). Here, a complete characterization of Banach space $$X=L_p(\mu)$$ having these properties is given: (a) $$X$$ has property (D) iff $$p\geq 2$$; (b) $$\, X$$ has property (dc) iff $$1<p\leq 2$$, and (c) $$X$$ has property (Cdc) iff $$p>1$$. The paper ends with some remarks and open problems. Stefan Cobzaş (Cluj-Napoca)