Sebestyén moment problem: The multi-dimensional case

Popovici, Dan; Sebestyén, Zoltán

doi:10.1090/S0002-9939-03-07291-5

Sebestyén moment problem: The multi-dimensional case
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by Dan Popovici and Zoltán Sebestyén PDF

Proc. Amer. Math. Soc. 132 (2004), 1029-1035 Request permission

Abstract:

Given a family $\{h_{\mathbf {n}}\}_{\mathbf {n}\in \mathbb {Z}_+^\Omega }$ of vectors in a Hilbert space $\mathcal {H}$ we characterize the existence of a family of commuting contractions $\mathbf {T}=\{T_\omega \}_{w\in \Omega }$ on $\mathcal {H}$ having regular dilation and such that \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 $|\Omega |=1$ or, recently, to Găvruţă and Păunescu in case $|\Omega |=2$.

References

S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961), 106–111 (German). MR 131169
P. Gãvruţã and D. Pãunescu, Sebestyén’s moment problem and regular dilations, Acta Math. Hungar. 94 (2002), no. 3, 223–232. MR 1905727, DOI 10.1023/A:1015627209095
Israel Halperin, Sz.-Nagy-Brehmer dilations, Acta Sci. Math. (Szeged) 23 (1962), 279–289. MR 156196
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Zoltán Sebestyén, Moment theorems for operators of Hilbert space, Acta Sci. Math. (Szeged) 44 (1982), no. 1-2, 165–171. MR 660523
Béla Sz.-Nagy, Bemerkungen zur vorstehenden Arbeit des Herrn S. Brehmer, Acta Sci. Math. (Szeged) 22 (1961), 112–114 (German). MR 131170
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190