We introduce a matricial approach to the truncated complex moment problem, and apply it to the case of moment matrices of flat data type, for which the columns corresponding to the homogeneous monomials in z andz of highest degree can be written in terms of monomials of lower degree. Necessary and sufficient conditions for the existence and uniqueness of representing measures are obtained in terms of positivity and extension criteria for moment matrices. We discuss the connection between complex moment problems and the subnormal completion problem for 2-variable weighted shifts, and present in detail the construction of solutions for truncated complex moment problems associated with monomials of degrees one and two. Finally, we present generalizations to multivariable moment and completion problems.

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Abstract. We develop new computational tests for existence and uniqueness of representing measures µ in the Truncated Complex Moment Problem:We characterize the existence of finitely atomic representing measures in terms of positivity and extension properties of the moment matrix M (n)(γ) associated with γ ≡ γ (2n) : γ 00 , . . . , γ 0,2n , . . . , γ 2n,0 , γ 00 > 0 (Theorem 1.5). We study conditions for flat (i.e., rank-preserving) extensions M (n + 1) of M (n) ≥ 0; each such extension corresponds to a distinct rank M (n)-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, we reduce the existence problem for minimal representing measures to the solubility of small systems of multivariable algebraic equations (Theorem 2.7). In a variety of applications, including cases of the quartic moment problem (n = 2; Theorem 1.10), we apply these tests so as to construct flat extensions and minimal representing measures. In other examples, we use these tests to demonstrate the non-existence of representing measures or the non-existence of minimal representing measures.Key words and phrases. Truncated complex moment problem, moment matrix extension block, flat extensions of positive matrices, recursively generated relations, algebraic variety of a moment sequence. Contents; γ is called a truncated moment sequence (of order 2n) and µ is called a representing measure for γ.In the present paper we provide new necessary or sufficient conditions for the existence of representing measures, particularly those which are minimal in the sense of being finitely atomic with the fewest atoms possible. In a variety of examples, including the so-called quartic moment problem (Theorem 1.10), we explicitly construct minimal representing measures, or we establish the existence of minimal representing measures via the theory of flat extensions of moment matrices (Theorem 2.7). We also identify new computational tests that can be used to prove the non-existence of representing measures, or the non-existence of finitely atomic representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems, that do not appear in the classical one-dimensional moment problem. We exhibit the failure of recursive consistency in Example 2.2, and we demonstrate the absence of normal consistency in Example 2.5; either phenomenon implies the non-existence of finitely atomic representing measures. In Examples 4.4-4.7 we illustrate the variety obstruction to the existence of any representing measure. All of our results (both positive and negative) indicate very explicitly why multivariable moment problems are so intractable; even in the positive cases, the existence of minimal measures for TCMP reduces to the solubility of small systems of multivariable algebraic equations, sy...

Abstract. Let γ ≡ γ (2n) denote a sequence of complex numbers γ 00 , γ 01 , γ 10 , . . . , γ 0,2n , . . . , γ 2n,0 (γ 00 > 0, γ ij =γ ji ), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel mea-, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mp i . We prove that there exists a rank M (n)-atomic representing measure for γ (2n) supported in K P if and only if M (n) ≥ 0 and there is some rankpreserving extension M (n + 1) for which Mp i (n + k i ) ≥ 0, where deg p i = 2k i or 2k i − 1 (1 ≤ i ≤ m).