Embry truncated complex moment problem

Let T be a cyclic subnormal operator on a Hilbert space ℋ with cyclic vector x₀ and let γ_ij:=(T*ⁱT ^jx₀,x₀), for any i,j ∈ ℕ ∪ {0}. The Bram-Halmos' characterization for subnormality of T involved a moment matrix M(n). In a parallel approach, we construct a moment matrix E(n) corresponding to Embry's characterization for subnormality of T. We discuss the relationship between M(n) and E(n) via the full moment problem. Next, given a collection of complex numbers γ≡{γ_ij} (0 ≤ i + j ≤ 2n, |i-j| ≤ n) with γ₀₀ > 0 and γ _ji = γ̄_ij, we consider the truncated complex moment problem for γ; this entails finding a positive Borel measure μ supported in the complex plane ℂ such that γ_ij = ∫z̄ⁱz^jdμ(z). We show that this moment problem can be solved when E(n) ≥ 0 and E(n) admits a flat extension E(n + k), where k = 1 when n is odd and k = 2 when n is even.