Embry truncated complex moment problem. (Q1414697)

The truncated complex moment problem consists in the following: given a collection of complex numbers \(\{\gamma_{ij}\}\) \((0\leq i+j\leq n)\), find a positive Borel measure \(\mu\) on \({\mathbb C}\) such that \(\gamma_{ij}=\int\bar z^iz^j \,d\mu(z)\). A criterion for solvability the truncated complex moment problem was given by \textit{R. Curto} and \textit{L. Fialkow} [Mem. Am. Math. Soc. 648 (1998; Zbl 0913.47016)] in terms of the corresponding moment matrix \(M(n)\). The authors consider the truncated complex moment problem with the moments \(\gamma_{ij}\) subject to the following restrictions \(0\leq i+j\leq n\), \(| i-j| \leq n\) and give a solvability criterion in terms of a submatrix \(E(n)\) of the matrix \(M(n)\). The proof is based on a generalization of the Halmos-Bram criterion for subnormality found by \textit{M. Embry} [Acta Sci. Math. 35, 61--64 (1973; Zbl 0263.47023)].