The full infinite dimensional moment problem on symmetric algebras of locally convex real spaces

This talk aims to introduce an infinite dimensional version of the classical full moment problem and explore certain instances which actually arise in several applied fields. The general theoretical question addressed is whether a linear functional $L$ on the symmetric algebra $S(V)$ of a locally convex topological real vector space $V$ can be represented as an integral w.r.t.\! a non-negative Radon measure supported on a fixed subset of the algebraic dual $V^*$ of $V$. I present a recent joint work with M. Ghasemi, S. Kuhlmann and M. Marshall where we get representations of continuous positive semidefinite linear functionals $L: S(V)\rightarrow \mathbb{R}$ as integrals w.r.t.\! uniquely determined Radon measures supported in special sorts of closed balls in the topological dual space $V\u27$ of $V$. A better characterization of the support is obtained when $L$ is positive on a $2d-$power module of $S(V)$. I compare these results with the corresponding ones for the full moment problem on locally convex nuclear spaces, pointing out the crucial roles played by the continuity and the quasi-analyticity assumptions on $L$ in determining the support of the representing measure. In particular, I focus on a joint work with T. Kuna and A. Rota where we derive an analogous result for functionals on the symmetric algebra of the space of test functions on $\mathbb{R}^d$ which are positive on quadratic modules but not necessarily continuous. This setting is indeed general enough to encompass many spaces which occur in concrete applications, e.g.\! the space of point configurations

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

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 $L_p(\mu)$ 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.Comment: 19 page

On the spherical convexity of quadratic functions

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cones are given