Integral Equations and Operator Theory

, Volume 42, Issue 4, pp 373–384

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

  • Călin-Grigore Ambrozie
  • Dan Timotin
Article

Abstract

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.

2000 Mathematics Subject Classification

47B32 47A20 47A57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ag1] J. Agler, The Arveson extension theorem and coanalytic models,Integral Equations Operator Theory 5 (1982), 608–631.Google Scholar
  2. [Ag2] J. Agler, Interpolation, preprint.Google Scholar
  3. [AgM] J. Agler and J.E. McCarthy, Nevanlinna-Pick kernels and localization, inOperator Theoretical Methods, ed. by A. Gheondea, R.N. Gologan and D. Timotin, The Theta Foundation, Bucharest, 2000, 1–20.Google Scholar
  4. [AS] O. P. Agrawal and N. Salinas, Sharp kernels and canonical subspaces,Amer. Jour. of Math. 110 (1988), 23–48.Google Scholar
  5. [AEM] C.G. Ambrozie, M. Engliš and V. Müller, Analytic models over general domains in Cn,J. Operator Theory, to appear.Google Scholar
  6. [Aro] N. Aronszajn, Theory of reproducing kernels,Trans. Amer. Math. Soc. 68 (1950), 337–404.Google Scholar
  7. [Arv] W.B. Arveson, Subalgebras of C*-algebras III, Multivariable operator theory,Acta Mathematica 181 (1998), 159–228.Google Scholar
  8. [At] A. Athavale, Holomorphic kernels and commuting operators,Trans. Amer. Math. Soc. 304 (1987), 101–110.Google Scholar
  9. [BB] F. Beatrous and J. Burbea, Positive-definiteness and its applications to interpolation problems for holomorphic functions,Trans. Amer. Math. Soc.,284 (1984), 247–270.Google Scholar
  10. [BTV] J.A. Ball, T.T. Trent and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, inOperator Theory and Analysis: The M.A. Kaashoek Anniversary Volume (Workshop in Amsterdam, Nov. 1997), Operator Theory: Advances and Applications Vol. 122, Birkhäuser-Verlag, 2001, 89–138.Google Scholar
  11. [CV] R. Curto, F.-H. Vasilescu: Standard operator models in the polydisc,Indiana Univ. Math. J. 42 (1993), 791–810.Google Scholar
  12. [McC] S. McCullough: The local De Branges-Rovnyak construction and complete Nevanlinna-Pick kernels, inAlgebraic Methods in Operator Theory, ed. by Raul Curto and Palle E.T. Jorgensen, Birkhauser Boston, 1994, 15–24.Google Scholar
  13. [MV] V. Müller and F.-H. Vasilescu: Standard models for some commuting multioperators,Proc. Amer. Math. Soc.,117 (1993), 979–989.Google Scholar
  14. [P1] G. Popescu, Interpolation problems in several variables,J. Math. Anal. Appl. 227 (1998), 227–250.Google Scholar
  15. [P2] G. Popescu, Commutant lifting, tensor algebras, and functional calculus,Proceedings of tie Edinburgh Mathematical Society, to appearGoogle Scholar
  16. [Po] S. Pott, Standard models under polynomial positivity conditions,J. Operator Theory 41 (1999), 365–389.Google Scholar
  17. [Q] P. Quiggin, For which reproducing kernel Hilbert spaces is Pick's theorem true?,Integral Equations Operator Theory 16 (1993), 244–266.Google Scholar
  18. [Sa] S. Saitoh,Theory of Reproducing Kernels and its Applications, Pitman Research Notes in math. no 189, Longman, 1988.Google Scholar
  19. [S] D. Sarason, Generalized interpolation inH ,Trans. Am. Math. Soc. 127 (1967), 2, 179–203.Google Scholar
  20. [SNF] B. Sz.-Nagy, C. Foias,Harmonic Analysis of Operators on Hilbert Space, Budapest, 1980.Google Scholar

Copyright information

© Birkhäuser Verlag 2002

Authors and Affiliations

  • Călin-Grigore Ambrozie
    • 1
  • Dan Timotin
    • 1
  1. 1.Institute of Mathematics of the Romanian AcademyBucharestRomania

Personalised recommendations