Sebestyén moment problem: The multi-dimensional case

Given a family { h n } n ∈ Z + Ω \{h_{\mathbf {n}}\}_{\mathbf {n}\in \mathbb {Z}_+^\Omega } of vectors in a Hilbert space H \mathcal {H} we characterize the existence of a family of commuting contractions T = { T ω } w ∈ Ω \mathbf {T}=\{T_\omega \}_{w\in \Omega } on H \mathcal {H} having regular dilation and such that h n = T n h 0 , n ∈ Z + Ω . \begin{equation*} h_{\mathbf {n}}=\mathbf {T} ^{\mathbf {n}} h_{\mathbf {0}},\quad \mathbf {n}\in \mathbb {Z}_+^\Omega . \end{equation*} The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case | Ω | = 1 |\Omega |=1 or, recently, to Găvruţă and Păunescu in case | Ω | = 2 |\Omega |=2 .