Skip to main content
SpringerLink
Log in

Bi-dimensional Moment Problems and Regular Dilations

  • Chapter
Operator Theory and Indefinite Inner Product Spaces

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 163))

Abstract

Given a map f on ℤ 2+ × X (X is just a non-empty set) into a Hilbert space ℌ we provide necessary and sufficient conditions in order to ensure the existence of a commuting pair (S, T) of contractions on ℌ having regular dilation such that

$$ S^m T^n f\left( {0,0,x} \right) = f\left( {m,n,x} \right), \left( {m,n} \right) \in \mathbb{Z}_ + ^2 , x \in X. $$

Such moment problems are strongly related to the theory of harmonizable and stationary processes. Isometric or unitary solutions are also characterized in terms of the initial data.

To Professor Heinz Langer

This work was supported by the EU Research Training Network “Analysis and Operators” with contract no. HPRN-CT-2000-00116.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Brehmer, Über vertauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. 22 (1961), 106–111.

    MATH  MathSciNet  Google Scholar 

  2. R.E. Curto, F.H. Vasilescu, Standard operator models in the polydisc. Indiana Univ. Math. J. 42 (1993), 791–810.

    Article  MathSciNet  MATH  Google Scholar 

  3. R.E. Curto, F.H. Vasilescu, Standard operator models in the polydisc II. Indiana Univ. Math. J. 44 (1995), 727–746.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Foiaş, A.E. Frazho, I. Gohberg, M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems. Operator Theory: Advances and Applications 100, Birkhäuser-Verlag, Basel, 1998.

    Google Scholar 

  5. I. Halperin, Sz.-Nagy-Brehmer dilations. Acta Sci. Math. 23 (1962), 279–289.

    MATH  MathSciNet  Google Scholar 

  6. Z.J. Jabłoński, J. Stochel, Subnormality and operator multidimensional moment problems. J. London Math. Soc. 71 (2005), 438–466.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Kakihara, Multidimensional second order stochastic processes. Series on Multivariate Analysis 2, World Scientific Publ. Co., Inc., River Edge, NJ, 1997.

    Google Scholar 

  8. A. Olofsson, Operator valued n-harmonic measure in the polydisc. Studia Math. 163 (2004), pp. 203–216.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Popovici, Matrix representations for regular dilations. Preprint (2004).

    Google Scholar 

  10. D. Popovici, On the unitary extension for a system of commuting isometries. Preprint (2004).

    Google Scholar 

  11. D. Popovici, Regular dilations on Kreřn spaces. Preprint (2004).

    Google Scholar 

  12. D. Popovici, Z. Sebestyén, Positive definite functions and Sebestyén’s operator moment problem. Glasgow Math. J. 47 (2005), 471–488.

    Article  MATH  Google Scholar 

  13. D. Popovici, Z. Sebestyén, Sebestyén moment problem: The multidimensional case. Proc. Amer. Math. Soc. 132 (2004), 1029–1035.

    Article  MathSciNet  MATH  Google Scholar 

  14. Z. Sebestyén, Moment theorems for operators on Hilbert space. Acta Sci. Math. 44 (1982), 165–171.

    MATH  Google Scholar 

  15. Z. Sebestyén, Moment theorems for operators on Hilbert space II. Acta Sci. Math. 47 (1984), 101–106.

    MATH  Google Scholar 

  16. Z. Sebestyén, A moment theorem for contractions on Hilbert spaces. Acta Math. Hung. 47 (1986), 391–393.

    Article  MATH  Google Scholar 

  17. Z. Sebestyén, Moment problem for dilatable semigroups of operators. Acta Sci. Math. 45 (1983), 365–376.

    MATH  Google Scholar 

  18. Z. Sebestyén, Moment type theorems for *-semigroups of operators on Hilbert space I. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 26 (1983), 213–218.

    MATH  Google Scholar 

  19. B. Sz.-Nagy, Bemerkungen zur vorstehenden Arbeit des Herrn S. Brehmer. Acta Sci. Math. 22 (1961), 112–114.

    MATH  MathSciNet  Google Scholar 

  20. B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert. Acta Sci. Math. 15 (1953), 87–92.

    MATH  MathSciNet  Google Scholar 

  21. B. Sz.-Nagy, Transformations de l’espace de Hilbert, fonctions de type positif sur un groupe. Acta Sci. Math. 15 (1954), 104–114.

    MathSciNet  Google Scholar 

  22. B. Sz.-Nagy, C. Foiaş, Harmonic analysis of operators on Hilbert space. North-Holland, Amsterdam, 1970.

    Google Scholar 

  23. D. Timotin, Regular dilations and models for multi-contractions. Indiana Univ. Math. J. 47 (1998), 671–684.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of the West Timişoara, B-dul Vasile Pârvan 4, RO-300223, Timişoara, Romania

    Dan Popovici

Authors
  1. Dan Popovici
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, UK

    Matthias Langer

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10 / 101, 1040, Wien, Austria

    Annemarie Luger  & Harald Woracek  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Popovici, D. (2005). Bi-dimensional Moment Problems and Regular Dilations. In: Langer, M., Luger, A., Woracek, H. (eds) Operator Theory and Indefinite Inner Product Spaces. Operator Theory: Advances and Applications, vol 163. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7516-7_11

Download citation

Publish with us

Policies and ethics

Search

Navigation