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{displaymath}h_{\mathbf{n}}=\mathbf{T} ^{\mathbf{n}} h_{\mathbf{0}},\quad \mathbf{n}\in\mathbb{Z} _+^\Omega. \end{displaymath}

The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case $\vert\Omega\vert=1$ or, recently, to Gavruta and Paunescu in case $\vert\Omega\vert=2$.

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

  • 1. S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. Szeged 22 (1961), 106–111 (German). MR 0131169
  • 2. P. Gãvruţã and D. Pãunescu, Sebestyén’s moment problem and regular dilations, Acta Math. Hungar. 94 (2002), no. 3, 223–232. MR 1905727, 10.1023/A:1015627209095
  • 3. Israel Halperin, Sz.-Nagy-Brehmer dilations, Acta Sci. Math. (Szeged) 23 (1962), 279–289. MR 0156196
  • 4. M. Neumark, Positive definite operator functions on a commutative group, Bull. Acad. Sci. URSS Sér. Math. [Izvestia Akad. Nauk SSSR] 7 (1943), 237–244 (Russian, with English summary). MR 0010265
  • 5. Zoltán Sebestyén, Moment theorems for operators of Hilbert space, Acta Sci. Math. (Szeged) 44 (1982), no. 1-2, 165–171. MR 660523
  • 6. Béla Sz.-Nagy, Bemerkungen zur vorstehenden Arbeit des Herrn S. Brehmer, Acta Sci. Math. Szeged 22 (1961), 112–114 (German). MR 0131170
  • 7. Béla Sz.-Nagy and Ciprian Foiaș, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. 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: Sebesty\'en 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