MathLAB Journal

Extension of linear operators and polynomial approximation, with applications to Markov moment problem and Mazur-Orlicz theorem

Abstract

Octav Olteanu

One recalls the relationship between the Markov moment problem and extension of linear functionals (or
operators), with two constraints. One states necessary and sufficient conditions for the existence of solutions of
some abstract vector-valued Markov moment problems, by means of a general Hahn-Banach principle. The
classical moment problem is discussed as a particular important case. This is the first aim of this review article
(see sections 1 and 2). Secondly short subsection (namely subsection 3.1) is devoted to applications of
polynomial approximation in studying the existence and uniqueness of the solutions for two types of Markov
moment problems. We use these general type results in studying related problems which involve concrete
spaces of functions and self-adjoint operators (subsection 3.2). This is the third purpose of the paper. Sometimes,
the uniqueness of the solution follows too. Most of our solutions are operator-valued or function-valued. The
methods follow from the corresponding proofs or via references citations. All the results have been previously
published (see the references mentioned in the beginning of each subsection or before the statements of the
theorems).<