Rational inner functions in the Schur-Agler class of the polydisk. (English) Zbl 1219.47028

Summary: Every two-variable rational inner function on the bidisk has a special representation called a unitary transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a unitary realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.


47A57 Linear operator methods in interpolation, moment and extension problems
42B05 Fourier series and coefficients in several variables
32A30 Other generalizations of function theory of one complex variable
32A70 Functional analysis techniques applied to functions of several complex variables
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[1] J. Agler, Some interpolation theorems of Nevanlinna-Pick type, unpublished manuscript (1988).
[2] J. Agler and J. E. M\mcc Carthy, “Pick interpolation and Hilbert function spaces” , Graduate Studies in Mathematics 44 , American Mathematical Society, Providence, RI, 2002. · Zbl 1010.47001
[3] T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88\Ndash90. · Zbl 0116.32403
[4] J. A. Ball, Multidimensional circuit synthesis and multivariable dilation theory, Multidim Syst Sign Process (to appear). · Zbl 1219.47025 · doi:10.1007/s11045-010-0123-2
[5] J. A. Ball and V. Bolotnikov, Canonical de Branges-Rovnyak model transfer-function realization for multivariable Schur-class functions, in: “Hilbert spaces of analytic functions” , CRM Proc. Lecture Notes 51 , Amer. Math. Soc., Providence, RI, 2010, pp. 1\Ndash39. · Zbl 1210.47041
[6] J. A. Ball, C. Sadosky, and V. Vinnikov, Scattering systems with several evolutions and multidimensional input/state/ output systems, Integral Equations Operator Theory 52(3) (2005), 323\Ndash393. · Zbl 1092.47006 · doi:10.1007/s00020-005-1351-y
[7] J. A. Ball and T. T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157(1) (1998), 1\Ndash61. · Zbl 0914.47020 · doi:10.1006/jfan.1998.3278
[8] B. J. Cole and J. Wermer, Ando’s theorem and sums of squares, Indiana Univ. Math. J. 48(3) (1999), 767\Ndash791. · Zbl 0945.47010 · doi:10.1512/iumj.1999.48.1716
[9] M. J. Crabb and A. M. Davie, Von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49\Ndash50. · Zbl 0301.47007 · doi:10.1112/blms/7.1.49
[10] G. Knese, Bernstein-Szegö measures on the two dimensional torus, Indiana Univ. Math. J. 57(3) (2008), 1353\Ndash1376. · Zbl 1268.42045 · doi:10.1512/iumj.2008.57.3226
[11] A. Kummert, Synthesis of two-dimensional lossless \(m\)-ports with prescribed scattering matrix, Circuits Systems Signal Process. 8(1) (1989), 97\Ndash119. · Zbl 0672.94027 · doi:10.1007/BF01598747
[12] W. Rudin, “Function theory in polydiscs” , W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0177.34101
[13] N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83\Ndash100. · Zbl 0288.47006 · doi:10.1016/0022-1236(74)90071-8
[14] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258\Ndash281. · Zbl 0042.12301 · doi:10.1002/mana.3210040124
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