Abstract

Let T be a cyclic subnormal operator on a Hilbert space H with cyclic vector x 0 and let γ ij :=( T * i T j x 0, x 0), 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⩽2 n, | i− j|⩽ n) with γ 00>0 and γ ji= γ ̄ 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=∫ z ̄ iz j dμ(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.

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