Moment problem for symmetric algebras of locally convex spaces

Abstract

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau }$ on the symmetric algebra $S(V)$. This allows the application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of $\overline{\tau }$-continuous linear functionals $L:S(V)\to \mathbb{R}$ satisfying $L(\sum S(V{)}^{2d})\subseteq [0,\infty )$ (more generally, $L(M)\subseteq [0,\infty )$ 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 Sifrin. It is more general because $V$ can be any lc 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 $L:S(V)\to \mathbb{R}$ is $\overline{\tau }$-continuous (not just continuous on each homogeneous part of $S(V)$).

19 pages, revised according to referee's comments, updated references, to appear in Integral Equations and Operator Theory

References in corpus (1)

• Infinite dimensional moment problem: open questions and applications

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