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St. Petersburg Mathematical Journal

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Operator-valued Bergman inner functions as transfer functions


Author: A. Olofsson
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 603-623
MSC (2000): Primary 47A48; Secondary 47A15
DOI: https://doi.org/10.1090/S1061-0022-08-01013-3
Published electronically: May 9, 2008
MathSciNet review: 2381937
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Abstract: An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discrete time linear systems. This points to a new interaction between the fields of invariant subspace theory and mathematical systems theory.


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  • 1. J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), 608-631. MR 0697007 (84g:47011)
  • 2. -, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 203-217. MR 0775993 (86i:47028)
  • 3. A. Aleman, S. Richter, and C. Sundberg, Beurling's theorem for the Bergman space, Acta Math. 177 (1996), 275-310. MR 1440934 (98a:46034)
  • 4. C.-G. Ambrozie, M. Engliš, and V. Müller, Operator tuples and analytic models over general domains in $ \mathbb{C}\sp n$, J. Operator Theory 47 (2002), 287-302. MR 1911848 (2004c:47013)
  • 5. J. Arazy and M. Engliš, Analytic models for commuting operator tuples on bounded symmetric domains, Trans. Amer. Math. Soc. 355 (2003), 837-864. MR 1932728 (2003k:47019)
  • 6. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • 7. J. A. Ball and N. Cohen, de Branges-Rovnyak operator models and systems theory: A survey, Topics in Matrix and Operator Theory (Rotterdam, 1989), Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 93-136. MR 1115026 (93a:47011)
  • 8. I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators. Vol. II, Oper. Theory Adv. Appl., vol. 63, Birkhäuser, Basel, 1993. MR 1246332 (95a:47001)
  • 9. P. R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125-134. MR 0044036 (13:359b)
  • 10. -, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102-112. MR 0152896 (27:2868)
  • 11. H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine Angew. Math. 422 (1991), 45-68. MR 1133317 (93c:30053)
  • 12. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. MR 1758653 (2001c:46043)
  • 13. J. W. Helton, Discrete time systems, operator models, and scattering theory, J. Funct. Anal. 16 (1974), 15-38. MR 0445310 (56:3652)
  • 14. A. Olofsson, Wandering subspace theorems, Integral Equations Operator Theory 51 (2005), 395-409. MR 2126818 (2005k:47019)
  • 15. -, An operator-valued Berezin transform and the class of $ n$-hypercontractions, Integral Equations Operator Theory 58 (2007), no. 4, 503-549. http://www.math.kth.se/~ao/. MR 2329133
  • 16. -, A characteristic operator function for the class of $ n$-hypercontractions, J. Funct. Anal. 236 (2006), 517-545. MR 2240173 (2007e:47018)
  • 17. S. M. Shimorin, Double power series and reproducing kernels, Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 339-348. MR 1771773 (2001d:46042)
  • 18. -, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147-189. MR 1810120 (2002c:47018)
  • 19. -, On Beurling-type theorems in weighted $ l\sp 2$ and Bergman spaces, Proc. Amer. Math. Soc. 131 (2003), 1777-1787. MR 1955265 (2004a:47008)
  • 20. B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland, New York, 1970. MR 0275190 (43:947)

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Additional Information

A. Olofsson
Affiliation: Falugatan 22 1tr, SE-113 32 Stockholm, Sweden
Email: ao@math.kth.se

DOI: https://doi.org/10.1090/S1061-0022-08-01013-3
Keywords: Bergman inner function, transfer function, $n$-hypercontraction, wandering subspace, standard weighted Bergman space, discrete time linear system
Received by editor(s): September 4, 2006
Published electronically: May 9, 2008
Additional Notes: Supported by the M.E.N.R.T. (France) and the G. S. Magnuson’s Fund of the Royal Swedish Academy of Sciences
Article copyright: © Copyright 2008 American Mathematical Society

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