arXiv subjects

Patrick Michalski

Publications and source records attributed to Patrick Michalski.

3 recordsLinked to original sources

arXiv2023

Moment problem for algebras generated by a nuclear space

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra $A$, which we assume to be generated by a vector space $V$ endowed with a Hilbertian seminorm $q$. Such a general criterion provides representing measures with support contained in the space of characters of $A$ whose restrictions to $V$ are $q-$continuous. This allows us in turn to prove existence results for the case when $V$ is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.

math.FA↗

arXiv2022

An intrinsic characterization of moment functionals in the compact case

We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is non-negative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.

math.FA↗

arXiv2022

Projective limits techniques for the infinite dimensional moment problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra $A$ be represented as an integral with respect to a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $σ-$algebra generated by the weak topology? We approach this problem by constructing $X(A)$ as a projective limit of the character spaces of all finitely generated unital subalgebras of $A$. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on $A$ with respect to a measure on the cylinder $σ-$algebra on $X(A)$ (resp. a Radon measure on the Borel $σ-$algebra on $X(A)$) provided that for any finitely generated unital subalgebra of $A$ the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on $A$. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of $A$. Our results in particular apply to the case when $A$ is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.

math.FA↗