Toral algebraic sets and function theory on polydisks
Cite this article as: Agler, J., McCarthy, J.E. & Stankus, M. J Geom Anal (2006) 16: 551. doi:10.1007/BF02922130 Abstract A toral algebraic set A is an algebraic set in ℂ n whose intersection with T n is 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
Communicated by John Erik Fornæss
Agler, J. and McCarthy, J.E. Nevanlinna-Pick interpolation on the bidisk,
J. Reine Angew. Math.
, 191–204, (1999).
Agler, J. and McCarthy, J.E. Distinguished varieties,
, 133–153, (2005).
Andô, T. On a pair of commutative contractions,
Acta Sci. Math. (Szeged)
, 88–90, (1963).
Ball, J.A., Sadosky, C., and Vinnikov, V. Conservative linear systems, unitary colligations and Lax-Phillips scattering: Multidimensional generalizations,
Internat. J. Control
(9), 802–811, (2004).
MATH CrossRef MathSciNet
Ball, J.A. and Trent, T.T. Unitary colligations, reproducing kernel Hubert spaces, and Nevanlinna-Pick interpolation in several variables,
J. Funct. Anal.
, 1–61, (1998).
Ball, J.A. and Vinnikov, V. Hardy spaces on a finite bordered Riemann surface, multivariable operator theory and Fourier analysis along a unimodular curve, in
Operator Theory Advances and Applications, 129, 37–56, Birkhäuser, Basel, (2000).
Ball, J.A. and Vinnikov, V. Overdetermined multidimensional systems: State space and frequency domain methods, in
Mathematical Systems Theory in Biology, Communications, Computation, and Finance, 134, IMA Vol. Math. Appl. 63–119, Springer, Berlin, (2003).
Cole, B.J. and Wermer, J. Andô’s theorem and sums of squares,
Indiana Math. J.
, 767–791, (1999).
MATH CrossRef MathSciNet
, Springer, New York, (1995).
Plane Algebraic Curves
, American Mathematical Society, Providence, (2001).
Krantz, S. and Parks, H.
A Primer of Real Analytic Functions
, Birkhäuser, Basel, (2002).
Function Theory in Polydiscs
, Benjamin, New York, (1969).
Szokefalvi-Nagy, B. and Foiaş, C.
Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, (1970). Copyright information
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