Moment Problem for Symmetric Algebras of Locally Convex Spaces

Abstract

It is explained how a locally convex (lc) topology τ on a real vector space V extends to a locally multiplicatively convex (lmc) topology τ 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 τ -continuous linear function-as integrals with respect to uniquely determined Radon measures µ 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 andŠifrin. 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 ) → R is τ -continuous (not just continuous on each homogeneous part of S(V )).Murray Marshall passed away in May 2015. He worked on this manuscript together with us until the very last days of his life. We lost a collaborator of many years and a wonderful friend. We sorely miss him. (M. Ghasemi, M. Infusino, S. Kuhlmann) 2010 Mathematics Subject Classification. Primary 44A60 Secondary 14P99.

Infinite dimensional moment problem: open questions and applications

“…In Section 3 we focus on the special case of linear functionals on the symmetric algebra of a real locally convex space, since this setting is general enough to include the classical finite dimensional moment problem and several of the infinite dimensional moment problems appearing in the applications mentioned above. More precisely, we will review the results of [14] and compare them with some previous results [2, Vol. II, Chapter 5, Section 2], [3], [8,Section 4], [22,Section 3], [25] about the moment problem on the symmetric algebra of a real locally convex (lc) space which is also nuclear.…”

“…In [14] it is indeed explained how a lc topology τ on a real vector space V can be extended to a lmc topology τ on the symmetric algebra S(V ) and a complete criterion for the existence of a solution to the K-moment problem forτ -continuous linear functionals on S(V ) is given. Furthermore, [14] contains a detailed comparison between these results and the ones in [2, Theorem 2.1], [3], [25,Theorem 2.3] for the K-moment problem on locally convex nuclear spaces. Starting from this comparison, we outline some connections to further previous works on the moment problem on symmetric algebras of lc nuclear spaces and pose some questions which naturally emerge from this analysis.…”

“…We start by briefly reporting the main results of [14] in the case V is a R−vector space with the topology given by a single seminorm ρ. The procedure to extend ρ on V to a submultiplicative seminorm ρ on S(V ) can be summarized as follows.…”

“…In [14] the authors consider the topology τ on S(V ) generated by the family of seminorms Q := {nρ : ρ ∈ P, n ∈ N}, which is directed in virtue of Remark 3.2. Hence, by using Proposition 2.5 and the universal property of the symmetric algebra, it is possible show the following result.…”

“…[1], [14], [18], [19], [25], [28], [29]). In particular, the results in [14] revealed intriguing relations between real algebraic geometry, functional and harmonic analysis, leading us to several open questions which we are going to present in this paper.…”