Operator-valued Bergman inner functions as transfer functions
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- by A. Olofsson
- St. Petersburg Math. J. 19 (2008), 603-623
- DOI: https://doi.org/10.1090/S1061-0022-08-01013-3
- Published electronically: May 9, 2008
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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.References
- Jim Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 5, 608–631. MR 697007, DOI 10.1007/BF01694057
- Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
- A. Aleman, S. Richter, and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996), no. 2, 275–310. MR 1440934, DOI 10.1007/BF02392623
- C.-G. Ambrozie, M. Engliš, and V. Müller, Operator tuples and analytic models over general domains in $\Bbb C^n$, J. Operator Theory 47 (2002), no. 2, 287–302. MR 1911848
- Jonathan Arazy and Miroslav Engliš, Analytic models for commuting operator tuples on bounded symmetric domains, Trans. Amer. Math. Soc. 355 (2003), no. 2, 837–864. MR 1932728, DOI 10.1090/S0002-9947-02-03156-2
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- Joseph A. Ball and Nir 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, DOI 10.1007/978-3-0348-5672-0_{5}
- Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, vol. 63, Birkhäuser Verlag, Basel, 1993. MR 1246332, DOI 10.1007/978-3-0348-8558-4_{1}
- Paul R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125–134. MR 44036
- Paul R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112. MR 152896, DOI 10.1515/crll.1961.208.102
- Håkan Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine Angew. Math. 422 (1991), 45–68. MR 1133317, DOI 10.1515/crll.1991.422.45
- Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
- J. William Helton, Discrete time systems, operator models, and scattering theory, J. Functional Analysis 16 (1974), 15–38. MR 0445310, DOI 10.1016/0022-1236(74)90069-x
- Anders Olofsson, Wandering subspace theorems, Integral Equations Operator Theory 51 (2005), no. 3, 395–409. MR 2126818, DOI 10.1007/s00020-003-1322-0
- Anders Olofsson, An operator-valued Berezin transform and the class of $n$-hypercontractions, Integral Equations Operator Theory 58 (2007), no. 4, 503–549. MR 2329133, DOI 10.1007/s00020-007-1502-4
- Anders Olofsson, A characteristic operator function for the class of $n$-hypercontractions, J. Funct. Anal. 236 (2006), no. 2, 517–545. MR 2240173, DOI 10.1016/j.jfa.2006.03.004
- Serguei 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
- Sergei Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147–189. MR 1810120, DOI 10.1515/crll.2001.013
- Serguei Shimorin, On Beurling-type theorems in weighted $l^2$ and Bergman spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1777–1787. MR 1955265, DOI 10.1090/S0002-9939-02-06721-7
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Bibliographic Information
- A. Olofsson
- Affiliation: Falugatan 22 1tr, SE-113 32 Stockholm, Sweden
- Email: ao@math.kth.se
- 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
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2381937