Embry truncated complex moment problem

AbstractLet T be a cyclic subnormal operator on a Hilbert space H with cyclic vector x0 and let γij:=(T*iTjx0,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≡{γij} (0⩽i+j⩽2n, |i−j|⩽n) with γ00>0 and γji=γ̄ij, 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 γij=∫z̄izjdμ(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