Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 877–915 | Cite as

A Conservative de Branges–Rovnyak Functional Model for Operator Schur Functions on \(\mathbb C^+\)

  • Joseph A. Ball
  • Mikael Kurula
  • Olof J. Staffans
Article
  • 51 Downloads

Abstract

We present a solution of the operator-valued Schur-function realization problem on the right-half plane by developing the corresponding de Branges–Rovnyak canonical conservative simple functional model. This model corresponds to the closely connected unitary model in the disk setting, but we work the theory out directly in the right-half plane, which allows us to exhibit structure which is absent in the disk case. A main feature of the study is that the connecting operator is unbounded, and so we need to make use of the theory of well-posed continuous-time systems. In order to strengthen the classical uniqueness result (which states uniqueness up to unitary similarity), we introduce non-invertible intertwinements of system nodes.

Keywords

Schur function Continuous time Linear system Right half-plane Functional model de Branges–Rovnyak Realization Reproducing kernel 

Mathematics Subject Classification

93B15 47A48 47B32 

References

  1. 1.
    Adamjan, V.M., Arov, D.Z.: Unitary couplings of semi-unitary operators, Mat. Issled. 1 (1966), no. vyp. 2, 3–64, (Russian) English translation available in Am. Math. Soc. Trans I. Ser. 2, 95 (1970), pp. 75–129Google Scholar
  2. 2.
    Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H.: Schur functions, operator colligations, and reproducing kernel Hilbert spaces. In: Gohberg, I. (ed.) Operator Theory: Advances and Applications, vol. 96. Birkhäuser, Basel. https://link.springer.com/content/pdf/bfm%3A978-3-0348-8908-7%2F1.pdf (1997)
  3. 3.
    Arov, D.Z., Kurula, M., Staffans, O.J.: Canonical state/signal shift realizations of passive continuous time behaviors. Complex Anal. Oper. Theory 5(2), 331–402 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arov, D.Z., Nudelman, M.A.: Passive linear stationary dynamical scattering systems with continuous time. Integral Equ. Oper. Theory 24, 1–45 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory. Part IV: affine representations of discrete time systems. Complex Anal. Oper. Theory 1, 457–521 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Arov, D.Z., Staffans, O.J.: Two canonical passive state/signal shift realizations of passive discrete time behaviors. J. Funct. Anal. 257, 2573–2634 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Arov, D.Z., Staffans, O.J.: Canonical conservative state/signal shift realizations of passive discrete time behaviors. J. Funct. Anal. 259(12), 3265–3327 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Behrndt, J., Hassi, S., de Snoo, H.: Functional models for Nevanlinna families. Opusc. Math. 28(3), 233–245 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Behrndt, J., Hassi, S., de Snoo, H.: Boundary relations, unitary colligations, and functional models. Complex Anal. Oper. Theory 3(1), 57–98 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ball, J.A., Kurula, M., Staffans, O.J., Zwart, H.: De Branges–Rovnyak realizations of operator-valued Schur functions on the complex right half-plane. Complex Anal. Oper. Theory 9(4), 723–792 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ball, J.A., Kaliuzhnyi-Verbovetskyi, D.S.: Schur–Agler and Herglotz–Agler classes of functions: positive-kernel decompositions and transfer-function realizations. Adv. Math. 280, 121–187 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brodskiĭ, M.S.: Unitary operator colligations and their characteristic functions. Russ. Math. Surv. 33(4), 159–191 (1978)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ball, J.A., Staffans, O.J.: Conservative state-space realizations of dissipative system behaviors. Integral Equ. Oper. Theory 54, 151–213 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    de Branges, L., Rovnyak, J.: Canonical models in quantum scattering theory. In: Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), pp. 295–392. Wiley, New York (1966)Google Scholar
  15. 15.
    de Branges, L., Rovnyak, J.: Square Summable Power Series. Holt, Rinehart and Winston, New York (1966)MATHGoogle Scholar
  16. 16.
    Nikolskiĭ, N.K., Vasyunin, V.I.: Notes on two function models, The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., vol. 21, pp. 113–141. American Mathematical Society, Providence, RI (1986)Google Scholar
  17. 17.
    Nikolskiĭ, N.K., Vasyunin, V.I.: A unified approach to function models, and the transcription problem. In: The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), Oper. Theory Adv. Appl., vol. 41, pp. 405–434. Birkhäuser, Basel (1989)Google Scholar
  18. 18.
    Nikolskiĭ, N.K., Vasyunin, V.I.: Elements of spectral theory in terms of the free function model. I. Basic constructions, Holomorphic spaces (Berkeley, CA, 1995), Math. Sci. Res. Inst. Publ., vol. 33, pp. 211–302. Cambridge Univ. Press, Cambridge (1998)Google Scholar
  19. 19.
    Rudin, W.: Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973)MATHGoogle Scholar
  20. 20.
    Staffans, O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005)CrossRefMATHGoogle Scholar
  21. 21.
    Staffans, O.J.: On scattering passive system nodes and maximal scattering dissipative operators. Proc. Am. Math. Soc. 141(4), 1377–1383 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • Mikael Kurula
    • 2
  • Olof J. Staffans
    • 2
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of Mathematics and StatisticsÅbo Akademi UniversityÅboFinland

Personalised recommendations