## Schur-Agler class rational inner functions on the tridisk

HTML articles powered by AMS MathViewer

- by Greg Knese PDF
- Proc. Amer. Math. Soc.
**139**(2011), 4063-4072 Request permission

## Abstract:

We prove two results with regard to rational inner functions in the Schur-Agler class of the tridisk. Every rational inner function of degree $(n,1,1)$ is in the Schur-Agler class, and every rational inner function of degree $(n,m,1)$ is in the Schur-Agler class after multiplication by a monomial of sufficiently high degree.## References

- 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**, DOI 10.1090/gsm/044 - J. Milne Anderson, Michael A. Dritschel, and James Rovnyak,
*Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball*, Comput. Methods Funct. Theory**8**(2008), no. 1-2, 339–361. MR**2419482**, DOI 10.1007/BF03321692 - Mihály Bakonyi and Geir Naevdal,
*On the matrix completion method for multidimensional moment problems*, Acta Sci. Math. (Szeged)**64**(1998), no. 3-4, 547–558. MR**1666043** - Joseph A. Ball and Vladimir Bolotnikov,
*A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur-Agler class*, J. Math. Anal. Appl.**273**(2002), no. 2, 328–348. MR**1932492**, DOI 10.1016/S0022-247X(02)00226-3 - Joseph A. Ball and Vladimir Bolotnikov,
*Canonical de Branges-Rovnyak model transfer-function realization for multivariable Schur-class functions*, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, Amer. Math. Soc., Providence, RI, 2010, pp. 1–39. MR**2648864**, DOI 10.1090/crmp/051/01 - Joseph A. Ball, Cora Sadosky, and Victor Vinnikov,
*Scattering systems with several evolutions and multidimensional input/state/output systems*, Integral Equations Operator Theory**52**(2005), no. 3, 323–393. MR**2184571**, DOI 10.1007/s00020-005-1351-y - Brian J. Cole and John Wermer,
*Ando’s theorem and sums of squares*, Indiana Univ. Math. J.**48**(1999), no. 3, 767–791. MR**1736979**, DOI 10.1512/iumj.1999.48.1716 - M. J. Crabb and A. M. Davie,
*von Neumann’s inequality for Hilbert space operators*, Bull. London Math. Soc.**7**(1975), 49–50. MR**365179**, DOI 10.1112/blms/7.1.49 - Michael A. Dritschel,
*On factorization of trigonometric polynomials*, Integral Equations Operator Theory**49**(2004), no. 1, 11–42. MR**2057766**, DOI 10.1007/s00020-002-1198-4 - Bogdan Dumitrescu,
*Positive trigonometric polynomials and signal processing applications*, Signals and Communication Technology, Springer, Dordrecht, 2007. MR**2309555** - Jean-Pierre Gabardo,
*Trigonometric moment problems for arbitrary finite subsets of $\textbf {Z}^n$*, Trans. Amer. Math. Soc.**350**(1998), no. 11, 4473–4498. MR**1443194**, DOI 10.1090/S0002-9947-98-02091-1 - Jeffrey S. Geronimo and Ming-Jun Lai,
*Factorization of multivariate positive Laurent polynomials*, J. Approx. Theory**139**(2006), no. 1-2, 327–345. MR**2220044**, DOI 10.1016/j.jat.2005.09.010 - Jeffrey S. Geronimo and Hugo J. Woerdeman,
*Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables*, Ann. of Math. (2)**160**(2004), no. 3, 839–906. MR**2144970**, DOI 10.4007/annals.2004.160.839 - John A. Holbrook,
*Schur norms and the multivariate von Neumann inequality*, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 375–386. MR**1902811** - Greg Knese,
*Bernstein-Szegő measures on the two dimensional torus*, Indiana Univ. Math. J.**57**(2008), no. 3, 1353–1376. MR**2429095**, DOI 10.1512/iumj.2008.57.3226 - Greg Knese,
*Polynomials with no zeros on the bidisk*, Anal. PDE**3**(2010), no. 2, 109–149. MR**2657451**, DOI 10.2140/apde.2010.3.109 - Knese, G. (2010b). Rational inner functions in the Schur-Agler class of the polydisk. To appear in Publicacions Matemàtiques.
- Knese, G. (2010c). Stable symmetric polynomials and the Schur-Agler class. Preprint.
- Anton Kummert,
*Synthesis of $3$-D lossless first-order one ports with lumped elements*, IEEE Trans. Circuits and Systems**36**(1989), no. 11, 1445–1449. MR**1020132**, DOI 10.1109/31.41302 - Anton Kummert,
*Synthesis of two-dimensional lossless $m$-ports with prescribed scattering matrix*, Circuits Systems Signal Process.**8**(1989), no. 1, 97–119. MR**998029**, DOI 10.1007/BF01598747 - B. A. Lotto,
*von Neumann’s inequality for commuting, diagonalizable contractions. I*, Proc. Amer. Math. Soc.**120**(1994), no. 3, 889–895. MR**1169881**, DOI 10.1090/S0002-9939-1994-1169881-8 - Megretski, A. (2003). Positivity of trigonometric polynomials. In
*Decision and Control, 2003. Proceedings of 42nd IEEE Conference on Decision and Control*, volume 4, IEEE, pages 3814 – 3817. - Daniel G. Quillen,
*On the representation of hermitian forms as sums of squares*, Invent. Math.**5**(1968), 237–242. MR**233770**, DOI 10.1007/BF01389773 - Walter Rudin,
*The extension problem for positive-definite functions*, Illinois J. Math.**7**(1963), 532–539. MR**151796** - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841** - 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), 83–100. MR**0355642**, DOI 10.1016/0022-1236(74)90071-8

## Additional Information

**Greg Knese**- Affiliation: Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350
- MR Author ID: 813491
- Email: geknese@bama.ua.edu
- Received by editor(s): October 5, 2010
- Published electronically: March 30, 2011
- Additional Notes: This research was supported by NSF grant DMS-1048775
- Communicated by: Richard Rochberg
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 4063-4072 - MSC (2010): Primary 47A57; Secondary 42B05
- DOI: https://doi.org/10.1090/S0002-9939-2011-10975-4
- MathSciNet review: 2823051