Toral algebraic sets and function theory on polydisks
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 Classifications14J70 32A65
Key Words and PhrasesToral variety inner function H∞ Pick interpolation
Unable to display preview. Download preview PDF.
- 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
- 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
- Szokefalvi-Nagy, B. and Foiaş, C.Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, (1970).Google Scholar