Moment problem for symmetric algebras of locally convex spaces

M. Ghasemi, M. Infusino, S. Kuhlmann, M. Marshall

Published 2015-07-24Version 1

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends naturally to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall in [J. Funct. Analysis, 266 no.2 (2014) 1041-1049] to obtain representations of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum S(V)^{2d}) \subseteq [0,\infty)$ (more generally, of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(M) \subseteq [0,\infty)$ for some $2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely determined Radon measures $\mu$ supported by special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and \v{S}ifrin in [Mathematical Physics and Applied Mathematics, 12, Kluwer Academic Publishers, 1995], [Ukrain. Mat. \v{Z}., 23 (1971) 291-306]. It is more general because $V$ can be any locally convex topological space (not just a separable nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that $L : S(V) \rightarrow \mathbb{R}$ is $\overline{\tau}$-continuous (not just that $L$ is continuous on the homogeneous parts of degree $k$ of $S(V)$, for each $k\ge 0$).