Integral Equations and Operator Theory

, Volume 40, Issue 2, pp 231–243 | Cite as

The Weiss conjecture on admissibility of observation operators for contraction semigroups

  • Birgit Jacob
  • Jonathan R. Partington
Article

Abstract

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that\(\parallel C\left( {sI - A} \right)^{ - 1} \parallel \leqslant M\sqrt {\operatorname{Re} s} \) for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.

MSC

Primary 93C15, 47D06 Secondary 47B35, 32A37 

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References

  1. [1]
    O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces.Illinois J. Math., 41(4):532–558 (1997).Google Scholar
  2. [2]
    F.F. Bonsall. Boundedness of Hankel matrices.J. London Math. Soc., (2), 29:289–300 (1984).Google Scholar
  3. [3]
    F.F. Bonsall. Condition for boundedness of Hankel matrices.Bull. London Math. Soc., 26:171–176 (1994).Google Scholar
  4. [4]
    E.B. Davies.One-Parameter Semigroups. London Math. Society Monographs vol. 15. Academic Press, London, 1980.Google Scholar
  5. [5]
    P. Grabowski and F. M. Callier. Admissible observation operators, semigroup criteria of admissibility.Integ. Equat. Operat. Theory, 25:182–198 (1996).Google Scholar
  6. [6]
    K. Hoffman.Banach Spaces of Analytic Functions. Prentice-Hall, 1962.Google Scholar
  7. [7]
    P. Koosis.Introduction to H p Spaces. Cambridge University Press, Cambridge, 1980.Google Scholar
  8. [8]
    Y. Meyer.Wavelets and operators. Cambridge University Press, Cambridge, 1992.Google Scholar
  9. [9]
    E.W. Packel. A semigroup analogue of Foguel's counterexample.Proc. Amer. Math. Soc., 21:240–244 (1969).Google Scholar
  10. [10]
    J.R. Partington and G. Weiss. Admissible observation operators for the right shift semigroup.Math. Control Signals Systems, 13:179–192 (2000).Google Scholar
  11. [11]
    A. Simard. Counterexamples concerning powers of sectorial operators on a Hilbert space.Bull. Austral. Math. Soc., 60:459–468 (1999).Google Scholar
  12. [12]
    B. Sz.-Nagy and C. Foiaş.Harmonic analysis of Operators on Hilbert Space. North-Holland Publishing Company, Amsterdam, London, 1970.Google Scholar
  13. [13]
    G. Weiss. Admissible observation operators for linear semigroups.Israel J. Math., 65:17–43 (1989).Google Scholar
  14. [14]
    G. Weiss. Two conjectures on the admissibility of control operators. In F. Kappel W. Desch, editor,Estimation and Control of Distributed Parameter Systems, pages 367–378, Basel, 1991. Birkhäuser Verlag.Google Scholar
  15. [15]
    G. Weiss. A powerful generalization of the Carleson measure theorem. In V. Blondel, E. Sontag, M. Vidyasagar, and J. Willems, editors,Open Problems in Mathematical Systems Theory and Control. Springer Verlag, 1998.Google Scholar

Copyright information

© Birkhäuser Verlag 2001

Authors and Affiliations

  • Birgit Jacob
    • 1
  • Jonathan R. Partington
    • 1
  1. 1.School of MathematicsUniversity of LeedsLeedsUK

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