The subnormal completion problem in several variables

The subnormal completion problem for d-variable weighted shifts is considered. Necessary and sufficient conditions are obtained for a collection of initial weights C={(α_γ⁽¹⁾,...,α_γ^(d))}γ∈Γ, where Γ is from a family of finite indexing sets which includes {γ∈N0d:0≤|γ|≤m}, to give rise to a d-variable subnormal weighted shift operator whose initial weights are given by C. The conditions are communicated in terms of a new solution of a corresponding truncated K-moment problem. The case when d = 2 and all cubic moments are known is investigated in detail. In particular, using the solution of the subnormal completion problem in d-variables presented here, an easily checked concrete sufficient condition is given for a solution to the cubic subnormal completion problem in 2-variables and also an example of a collection of weights in 2-variables with cubic moments is provided which satisfies natural positivity conditions yet does not admit a subnormal completion.