Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

Abstract

As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T1≤T≤T2 for the linear operator T, only in terms of quadratic forms, is deduced. Here, T1, T,and T2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples.