Pick interpolation in several variables

Author:
Ryan Hamilton

Journal:
Proc. Amer. Math. Soc. **141** (2013), 2097-2103

MSC (2010):
Primary 47A57; Secondary 30E05, 46E22

DOI:
https://doi.org/10.1090/S0002-9939-2013-11571-6

Published electronically:
January 30, 2013

MathSciNet review:
3034435

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in .

**1.**M. B. Abrahamse,*The Pick interpolation theorem for finitely connected domains*, Michigan Math. J.**26**(1979), no. 2, 195–203. MR**532320****2.**Jim Agler and John E. McCarthy,*Nevanlinna-Pick interpolation on the bidisk*, J. Reine Angew. Math.**506**(1999), 191–204. MR**1665697**, https://doi.org/10.1515/crll.1999.004**3.**Jim Agler and John E. McCarthy,*Pick interpolation and Hilbert function spaces*, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR**1882259****4.**Hari Bercovici,*The algebra of multiplication operators on Bergman spaces*, Arch. Math. (Basel)**48**(1987), no. 2, 165–174. MR**878428**, https://doi.org/10.1007/BF01189287**5.**Hari Bercovici, Ciprian Foias, and Carl Pearcy,*Dual algebras with applications to invariant subspaces and dilation theory*, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR**787041****6.**Hari Bercovici and Derek Westwood,*The factorization of functions in the polydisc*, Houston J. Math.**18**(1992), no. 1, 1–6. MR**1159434****7.**Brian Cole, Keith Lewis, and John Wermer,*Pick conditions on a uniform algebra and von Neumann inequalities*, J. Funct. Anal.**107**(1992), no. 2, 235–254. MR**1172022**, https://doi.org/10.1016/0022-1236(92)90105-R**8.**Kenneth R. Davidson and Ryan Hamilton,*Nevanlinna-Pick interpolation and factorization of linear functionals*, Integral Equations Operator Theory**70**(2011), no. 1, 125–149. MR**2786738**, https://doi.org/10.1007/s00020-011-1862-7**9.**Ronald G. Douglas and Jaydeb Sarkar,*Some remarks on the Toeplitz corona problem*, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, Amer. Math. Soc., Providence, RI, 2010, pp. 81–89. MR**2648868****10.**Steven G. Krantz,*Function theory of several complex variables*, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR**1162310****11.**Scott McCullough,*Nevanlinna-Pick type interpolation in a dual algebra*, J. Funct. Anal.**135**(1996), no. 1, 93–131. MR**1367626**, https://doi.org/10.1006/jfan.1996.0005**12.**B. Prunaru,*A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces*, Canadian Math. Bull., to appear.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
47A57,
30E05,
46E22

Retrieve articles in all journals with MSC (2010): 47A57, 30E05, 46E22

Additional Information

**Ryan Hamilton**

Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L–3G1, Canada

Email:
rhamilto@uwaterloo.ca

DOI:
https://doi.org/10.1090/S0002-9939-2013-11571-6

Keywords:
Pick interpolation,
reproducing kernel,
dual algebras

Received by editor(s):
October 5, 2011

Published electronically:
January 30, 2013

Additional Notes:
This work was partially supported by an NSERC graduate scholarship

Communicated by:
Richard Rochberg

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.