The Journal of Geometric Analysis

, Volume 16, Issue 4, pp 551–562 | Cite as

Toral algebraic sets and function theory on polydisks

  • Jim Agler
  • John E. McCarthy
  • Mark Stankus
Article

Abstract

A toral algebraic set A is an algebraic set innwhose intersection with Tnis sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.

Math Subject Classifications

14J70 32A65 

Key Words and Phrases

Toral variety inner function H Pick interpolation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agler, J. and McCarthy, J.E. Nevanlinna-Pick interpolation on the bidisk,J. Reine Angew. Math. 506, 191–204, (1999).MATHMathSciNetGoogle Scholar
  2. [2]
    Agler, J. and McCarthy, J.E. Distinguished varieties,Acta Math. 94, 133–153, (2005).CrossRefMathSciNetGoogle Scholar
  3. [3]
    Andô, T. On a pair of commutative contractions,Acta Sci. Math. (Szeged) 24, 88–90, (1963).MATHMathSciNetGoogle Scholar
  4. [4]
    Ball, J.A., Sadosky, C., and Vinnikov, V. Conservative linear systems, unitary colligations and Lax-Phillips scattering: Multidimensional generalizations,Internat. J. Control 77(9), 802–811, (2004).MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Ball, J.A. and Trent, T.T. Unitary colligations, reproducing kernel Hubert spaces, and Nevanlinna-Pick interpolation in several variables,J. Funct. Anal. 197, 1–61, (1998).CrossRefMathSciNetGoogle Scholar
  6. [6]
    Ball, J.A. and Vinnikov, V. Hardy spaces on a finite bordered Riemann surface, multivariable operator theory and Fourier analysis along a unimodular curve, inOperator Theory Advances and Applications,129, 37–56, Birkhäuser, Basel, (2000).Google Scholar
  7. [7]
    Ball, J.A. and Vinnikov, V. Overdetermined multidimensional systems: State space and frequency domain methods, inMathematical Systems Theory in Biology, Communications, Computation, and Finance,134,IMA Vol. Math. Appl. 63–119, Springer, Berlin, (2003).Google Scholar
  8. [8]
    Cole, B.J. and Wermer, J. Andô’s theorem and sums of squares,Indiana Math. J. 48, 767–791, (1999).MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Eisenbud, D.Commutative Algebra, Springer, New York, (1995).MATHGoogle Scholar
  10. [10]
    Fischer, G.Plane Algebraic Curves, American Mathematical Society, Providence, (2001).MATHGoogle Scholar
  11. [11]
    Krantz, S. and Parks, H.A Primer of Real Analytic Functions, Birkhäuser, Basel, (2002).MATHGoogle Scholar
  12. [12]
    Rudin, W.Function Theory in Polydiscs, Benjamin, New York, (1969).MATHGoogle Scholar
  13. [13]
    Szokefalvi-Nagy, B. and Foiaş, C.Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, (1970).Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2006

Authors and Affiliations

  • Jim Agler
    • 1
  • John E. McCarthy
    • 2
  • Mark Stankus
    • 3
  1. 1.U.C. San DiegoLa Jolla
  2. 2.Washington UniversitySt. Louis
  3. 3.California Polytechnic State UniversitySan Luis Obispo

Personalised recommendations