The truncated multidimensional moment problem: canonical solutions

For the truncated multidimensional moment problem we introduce a notion of a canonical solution. Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space. It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty). In the case of the two-dimensional moment problem (with triangular truncations) a search for canonical solutions leads to an algebraic system of equations. A notion of the index iₛ of nonself-adjointness for a set of prescribed moments is introduced. The case iₛ=0 corresponds to flatness. In the case iₛ=1 we get explicit necessary and sufficient conditions for the existence of canonical solutions. These conditions are valid for arbitrary sizes of truncations. In the case iₛ=2 we get either explicit conditions for the existence of canonical solutions or a single quadratic equation with several unknowns. Numerical examples are provided.