• Kyungpook Mathematical Journal
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• 제54권2호
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• pp.157-163
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• 2014

For a finite subset ${\Lambda}$ of $\mathbb{N}_0{\times}\mathbb{N}_0$, where $\mathbb{N}_0$ is the set of nonnegative integers, we say that a complex data ${\gamma}_{\Lambda}:=\{{\gamma}_{ij}\}_{(ij){\in}{\Lambda}}$ in the unit disc $\mathbf{D}$ of complex numbers has a cyclic subnormal completion if there exists a Hilbert space $\mathcal{H}$ and a cyclic subnormal operator S on $\mathcal{H}$ with a unit cyclic vector $x_0{\in}\mathcal{H}$ such that ${\langle}S^{*i}S^jx_0,x_0{\rangle}={\gamma}_{ij}$ for all $i,j{\in}\mathbb{N}_0$. In this note, we obtain some sufficient conditions for a cyclic subnormal completion of ${\gamma}_{\Lambda}$, where ${\Lambda}$ is a finite subset of $\mathbb{N}_0{\times}\mathbb{N}_0$.

• 대한수학회지
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• 제44권4호
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• pp.949-970
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• 2007

By considering a bridge between Bram-Halmos and Embry characterizations for the subnormality of cyclic operators, we extend the Curto-Fialkow and Embry truncated complex moment problem, and solve the problem finding the finitely atomic representing measure ${\mu}$ such that ${\gamma}_{ij}={\int}\bar{z}^iz^jd{\mu},\;(0{\le}i+j{\le}2n,\;|i-j|{\le}n+s,\;0{\le}s{\le}n);$ the cases of s = n and s = 0 are induced by Bram-Halmos and Embry characterizations, respectively. The former is the Curto-Fialkow truncated complex moment problem and the latter is the Embry truncated complex moment problem.