Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sebestyén moment problem: The multi-dimensional case

Authors: Dan Popovici and Zoltán Sebestyén
Journal: Proc. Amer. Math. Soc. 132 (2004), 1029-1035
MSC (2000): Primary 47A57, 47A20
Published electronically: December 1, 2003
MathSciNet review: 2045418
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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
  • Israel Halperin, Sz.-Nagy-Brehmer dilations, Acta Sci. Math. (Szeged) 23 (1962), 279–289. MR 156196
  • M. A. Neumark, Positive definite operator functions on a commutative group, Bulletin Acad. Sci. URSS Sér. Math. 7 (1943), 237-244.
  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A57, 47A20

Retrieve articles in all journals with MSC (2000): 47A57, 47A20

Additional Information

Dan Popovici
Affiliation: Department of Mathematics, University of the West, Ro-1900 Timişoara, Bd. V. Pârvan 4, Romania

Zoltán Sebestyén
Affiliation: Department of Applied Analysis Loránd Eötvös University, H-1117 Budapest, Pázmány Péter sétány 1/C, Hungary

Keywords: Sebestyén operator moment problem, multi-contraction, regular dilation, positive definite function
Received by editor(s): October 22, 2002
Published electronically: December 1, 2003
Dedicated: To the memory of Gyula Farkas
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society