The Complex Moment Problem

Abstract

In this chapter, we give a digression into the complex moment problem on \(\mathbb{C}^{d}\). In Sect. 15.1, we discuss the equivalence of the complex moment problem on \(\mathbb{C}^{d}\) and the real moment problem on \(\mathbb{R}^{2d}\). In Sect. 15.2, we briefly treat the moment problems for two important ∗-semigroups (\(\mathbb{Z}^{d}\) and \(\mathbb{N}_{0} \times \mathbb{Z}^{d}\)). The operator-theoretic approach to the complex moment problem (Theorem 15.6) is developed in Sect. 15.3. In Sect. 15.4, we show that each positive functional on \(\mathbb{C}_{d}[\underline{z},\overline{\underline{z}}]\) satisfying the complex multivariate Carleman condition is a determinate moment functional (Theorem 15.11). In Sect. 15.5, moment functionals on \(\mathbb{C}[z,\overline{z}]\) are characterized in terms of extensions to a larger algebra (Theorem 15.14). Section 15.6 solves the two-sided complex moment problem on the complex plane (Theorem 15.15).