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
j
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)