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Notes on interpolation in the generalized Schur class. II. Nudel$'$man's problem

Authors: D. Alpay, T. Constantinescu, A. Dijksma and J. Rovnyak
Journal: Trans. Amer. Math. Soc. 355 (2003), 813-836
MSC (2000): Primary 47A57, 30E05, 47B32; Secondary 47B50, 42A50
Published electronically: October 9, 2002
MathSciNet review: 1932727
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Abstract: An indefinite generalization of Nudel$'$man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Carathéodory-Fejér interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel$'$man's problem.

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  • 1. V. M. Adamjan, D. Z. Arov, and M. G. Kre{\u{\i}}\kern.15emn, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86 (128) (1971), 34-75. MR 45:7505
  • 2. D. Alpay, V. Bolotnikov, and A. Dijksma, On the Nevanlinna-Pick interpolation problem for generalized Stieltjes functions, Integral Equations Operator Theory 30 (1998), no. 4, 379-408. MR 99k:47032
  • 3. D. Alpay, V. Bolotnikov, A. Dijksma, and J. Rovnyak, Some extensions of Loewner's theory of monotone operator functions, J. Funct. Anal., to appear.
  • 4. D. Alpay, T. Constantinescu, A. Dijksma, and J. Rovnyak, Notes on interpolation in the generalized Schur class. I. Applications of realization theory, Oper. Theory Adv. Appl., Birkhäuser, Basel, to appear.
  • 5. D. Alpay, A. Dijksma, and H. Langer, On the Loewner problem in the class ${N}_\kappa$, Proc. Amer. Math. Soc. 130 (2002), 2057-2066.
  • 6. D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997. MR 2000a:47024
  • 7. D. Alpay and J. Rovnyak, Loewner's theorem for kernels having a finite number of negative squares, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1109-1117. MR 99m:47013
  • 8. A. Amirshadyan and V. Derkach, Interpolation in generalized Nevanlinna and Stieltjes classes, J. Operator Theory 42 (1999), no. 1, 145-188. MR 2000e:47026
  • 9. R. Arocena, T. Ya. Azizov, A. Dijksma, and S. A. M. Marcantognini, On commutant lifting with finite defect. II, J. Funct. Anal. 144 (1997), no. 1, 105-116. MR 98a:47006
  • 10. J. A. Ball, I. Gohberg, and L. Rodman, Interpolation of rational matrix functions, Oper. Theory Adv. Appl., vol. 45, Birkhäuser, Basel, 1990. MR 92m:47027
  • 11. J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group ${ {U}}(m,\,n)$ which contains most classical interpolation theory, J. Operator Theory 9 (1983), no. 1, 107-142. MR 84m:47046
  • 12. V. Bolotnikov and H. Dym, On degenerate interpolation, entropy and extremal problems for matrix Schur functions, Integral Equations Operator Theory 32 (1998), no. 4, 367-435. MR 99m:47014
  • 13. T. Constantinescu and A. Gheondea, On the indefinite trigonometric moment problem of I. S. Iohvidov and M. G. Kre{\u{\i}}\kern.15emn, Math. Nachr. 171 (1995), 79-94. MR 95m:47019
  • 14. -, On the Carathéodory type problem of M. G. Kre{\u{\i}}\kern.15emn and H. Langer, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 3, 243-247. MR 99i:47025
  • 15. L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966. MR 35:5909
  • 16. A. Dijksma and H. Langer, Notes on a Nevanlinna-Pick interpolation problem for generalized Nevanlinna functions, Topics in interpolation theory (Leipzig, 1994), Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 69-91. MR 98g:47015
  • 17. C. Foias, A. E. Frazho, and M. A. Kaashoek, A weighted version of almost commutant lifting, Systems, approximation, singular integral operators and related topics, Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 311-340.
  • 18. A. Gheondea, Contractive intertwining dilations of quasi-contractions, Z. Anal. Anwendungen 15 (1996), no. 1, 31-44. MR 97e:47030
  • 19. L. B. Golinski{\u{\i}}\kern.15em, A generalization of the matrix Nevanlinna-Pick problem, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 3, 187-205, English transl., Soviet J. Contemporary Math. Anal. 18 (1983), no. 3, 22-39; private English translation by V. Katsnelson. MR 85g:47049
  • 20. S. Hassi, H. de Snoo, and H. Woracek, Some interpolation problems of Nevanlinna-Pick type. The Kre{\u{\i}}\kern.15emn-Langer method, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., vol. 106, Birkhäuser, Basel, 1998, pp. 201-216. MR 2001c:47022
  • 21. J. W. Helton, Orbit structure of the Möbius transformation semigroup acting on ${H}\sp{\infty }$ (broadband matching), Topics in functional analysis (essays dedicated to M. G. Kre{\u{\i}}\kern.15emn on the occasion of his 70th birthday), Academic Press, New York, 1978, pp. 129-157. MR 81e:46019
  • 22. T. S. Ivanchenko, The Nevanlinna-Pick problem in the case of an indefinite metric, Dokl. Akad. Nauk Ukrain. SSR Ser. A (1980), no. 5, 8-13, 92. MR 84i:47056
  • 23. M. G. Kre{\u{\i}}\kern.15emn and H. Langer, Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume ${\Pi} \sb{\kappa }$, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, Amsterdam, 1972, pp. 353-399. Colloq. Math. Soc. János Bolyai, 5. MR 54:11103
  • 24. -, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ${\Pi} \sb{\kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236. MR 57:1173
  • 25. A. A. Nudel$'$man, A new problem of the type of the moment problem, Dokl. Akad. Nauk SSSR 233 (1977), no. 5, 792-795, English transl., Soviet Math. Dokl. 18 (1977), 507-510. MR 57:10379
  • 26. -, A generalization of classical interpolation problems, Dokl. Akad. Nauk SSSR 256 (1981), no. 4, 790-793. MR 82f:30033
  • 27. V. V. Peller, An excursion into the theory of Hankel operators, Holomorphic Spaces (S. Axler, J. E. McCarthy, and D. Sarason, eds.), MSRI Publications, vol. 33, Cambridge University Press, Cambridge, 1998, pp. 65-120. MR 99e:47033
  • 28. G. Popescu, Meromorphic interpolation in several variables, preprint, 2001.
  • 29. M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, Oxford University Press, New York, 1985, Dover republication, New York, 1997. MR 97j:47002
  • 30. -, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Basel, 1994. MR 97a:30047
  • 31. A. L. Sakhnovich, Modification of V. P. Potapov's scheme in the indefinite case, Matrix and operator valued functions, Oper. Theory Adv. Appl., vol. 72, Birkhäuser, Basel, 1994, pp. 185-201. MR 96b:47016
  • 32. D. Sarason, Generalized interpolation in ${H}\sp{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 34:8193
  • 33. -, Sub-Hardy Hilbert spaces in the unit disk, John Wiley & Sons Inc., New York, 1994. MR 96k:46039
  • 34. -, Nevanlinna-Pick interpolation with boundary data, Integral Equations Operator Theory 30 (1998), no. 2, 231-250. MR 98m:47017
  • 35. M. Schreiber, A functional calculus for general operators in Hilbert space, Trans. Amer. Math. Soc. 87 (1958), 108-118. MR 20:6040
  • 36. B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam, 1970. MR 43:947

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Additional Information

D. Alpay
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

T. Constantinescu
Affiliation: Programs in Mathematical Sciences, University of Texas at Dallas, Box 830688, Richardson, Texas 75083-0688

A. Dijksma
Affiliation: Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands

J. Rovnyak
Affiliation: University of Virginia, Department of Mathematics, P.O. Box 400137, Charlottesville, Virginia 22904-4137
Email: rovnyak@Virginia.EDU

Received by editor(s): September 18, 2001
Received by editor(s) in revised form: April 16, 2002
Published electronically: October 9, 2002
Additional Notes: J. Rovnyak was supported by the National Science Foundation DMS-0100437 and by the Netherlands Organization for Scientific Research NWO B 61-482.
Article copyright: © Copyright 2002 American Mathematical Society

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