Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Interpolation for multipliers on reproducing kernel Hilbert spaces

Author: Vladimir Bolotnikov
Journal: Proc. Amer. Math. Soc. 131 (2003), 1373-1383
MSC (2000): Primary 41A05, 46E22
Published electronically: December 6, 2002
MathSciNet review: 1949867
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: All solutions of a tangential interpolation problem for contractive multipliers between two reproducing kernel Hilbert spaces of analytic vector-valued functions are characterized in terms of certain positive kernels. In a special important case when the spaces consist of analytic functions on the unit ball of $\mathbb{C} ^d$ and the reproducing kernels are of the form $(1-\langle z,w\rangle^{-1})I_p$ and $(1-\langle z,w\rangle)^{-1}I_q$, the characterization leads to a parametrization of the set of all solutions in terms of a linear fractional transformation.

References [Enhancements On Off] (What's this?)

  • 1. J. Agler and J. E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal., 175 (2000), 111-124. MR 2001h:47019
  • 2. D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert modules and applications, in: Topics in Interpolation Theory (H. Dym, B. Fritzsche, V. Katsnelson and B. Kirstein, eds.), Oper. Theory Adv. Appl., OT95, Birkhäuser Verlag, Basel, 1997, pp. 37-68. MR 99b:30055
  • 3. D. Alpay, V. Bolotnikov and H. T. Kaptanoglu, The Schur algorithm and reproducing kernel Hilbert spaces in the ball, Linear Algebra Appl., 342 (2002), 163-186. MR 2002m:47019
  • 4. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404. MR 14:479c
  • 5. W. Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159-228. MR 2000e:47013
  • 6. J. A. Ball, T. T. Trent and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernels Hilbert spaces, Oper. Theory Adv. Appl. 122 (2001), 89-138. MR 2002f:47028
  • 7. V. Bolotnikov and H. Dym, On boundary interpolation for matrix Schur functions, Preprint, 1999.
  • 8. L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966. MR 35:5909
  • 9. F. Beatrous and J. Burbea, Positive-definiteness and its applications to interpolation problems for holomorphic functions, Trans. Amer. Math. Soc., 284 (1984), no.1, 247-270. MR 85e:32020
  • 10. H. Dym, $J$ contractive matrix functions, reproducing kernel spaces and interpolation, CBMS Lecture Notes, vol. 71, Amer. Math. Soc., Rhode Island, 1989. MR 90g:47003
  • 11. I. V. Kovalishina and V. P. Potapov, Seven Papers Translated from the Russian, Amer. Math. Soc. Transl. (2), 138, Providence, R.I., 1988. MR 89f:00030
  • 12. S. McCullough, The local de Branges-Rovnyak construction and complete Nevanlinna-Pick kernels, in Algebraic methods in operator theory (Ed. R. Curto and P. E. T. Jorgensen), Birkhäuser-Verlag, Boston, 1994, pp. 15-24. MR 95j:47016
  • 13. G. Popescu, Interpolation problems in several variables, J. Math. Anal. Appl., 227 (1998), 227-250. MR 99i:47028
  • 14. P. Quiggin, For which reproducing kernel Hilbert spaces is Pick's theorem true?, Integral Equations Operator Theory 16 (1993), no. 2, 244-266. MR 94a:47026
  • 15. D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, John Wiley and Sons Inc., New York, 1994. MR 96k:46039

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A05, 46E22

Retrieve articles in all journals with MSC (2000): 41A05, 46E22

Additional Information

Vladimir Bolotnikov
Affiliation: Department of Mathematics, The College of William and Mary, Williamsburg, Virginia 23187-8795

Received by editor(s): February 24, 2001
Received by editor(s) in revised form: March 23, 2001
Published electronically: December 6, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society