Outer factorizations in one and several variables

Dritschel, Michael; Woerdeman, Hugo

doi:10.1090/S0002-9947-05-03814-6

Outer factorizations in one and several variables

Authors: Michael A. Dritschel and Hugo J. Woerdeman
Journal: Trans. Amer. Math. Soc. 357 (2005), 4661-4679
MSC (2000): Primary 47A68, 47B35, 15A48
DOI: https://doi.org/10.1090/S0002-9947-05-03814-6
Published electronically: June 21, 2005
MathSciNet review: 2156726
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Abstract: A multivariate version of Rosenblum's Fejér-Riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered. Due to a simplification of the proof of the single variable case, new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved.

1. Gr. Arsene, Zoia Ceauşescu, and T. Constantinescu, Schur analysis of some completion problems, Linear Algebra Appl. 109 (1988), 1–35. MR 961563, https://doi.org/10.1016/0024-3795(88)90195-4
2. Mihály Bakonyi and Hugo J. Woerdeman, The central method for positive semi-definite, contractive and strong Parrott type completion problems, Operator theory and complex analysis (Sapporo, 1991) Oper. Theory Adv. Appl., vol. 59, Birkhäuser, Basel, 1992, pp. 78–95. MR 1246810
3. A. P. Calderón and R. Pepinsky.
On the phases of Fourier coefficients for positive real periodic functions.
In Ray Pepinsky, editor, Computing methods and the phase problem in -ray crystal analysis, pages 339-348. The X-Ray Crystal Analysis Laboratory, Department of Physics, The Pennsylvannia State College, 1952.
4. Raymond Cheng, Weakly and strongly outer functions on the bidisc, Michigan Math. J. 39 (1992), no. 1, 99–109. MR 1137892, https://doi.org/10.1307/mmj/1029004458
5. M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials, 𝐾-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 103–126. MR 1327293
6. T. Constantinescu, Schur analysis of positive block-matrices, I. Schur methods in operator theory and signal processing, Oper. Theory Adv. Appl., vol. 18, Birkhäuser, Basel, 1986, pp. 191–206. MR 902605, https://doi.org/10.1007/978-3-0348-5483-2_7
7. Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
8. Michael A. Dritschel, On factorization of trigonometric polynomials, Integral Equations Operator Theory 49 (2004), no. 1, 11–42. MR 2057766, https://doi.org/10.1007/s00020-002-1198-4
9. Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546
10. Jeffrey S. Geronimo and Hugo J. Woerdeman.
Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables.
Ann. of Math. (2), 160:839-906, 2004.
11. I. C. Gohberg, The factorization problem for operator functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1055–1082 (Russian). MR 0174994
12. Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
13. J. W. McLean and H. J. Woerdeman, Spectral factorizations and sums of squares representations via semidefinite programming, SIAM J. Matrix Anal. Appl. 23 (2001/02), no. 3, 646–655. MR 1896811, https://doi.org/10.1137/S0895479800371177
14. Pablo A. Parrilo.
Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization.
Ph.D. thesis, California Institute of Technology, 2000.
15. Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. MR 1068530
16. Murray Rosenblatt, A multi-dimensional prediction problem, Ark. Mat. 3 (1958), 407–424. MR 92332, https://doi.org/10.1007/BF02589495
17. Marvin Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139–147. MR 227794, https://doi.org/10.1016/0022-247X(68)90122-4
18. Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228
19. Walter Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532–539. MR 0151796

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Additional Information

Michael A. Dritschel
Affiliation: School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, United Kingdom
Email: m.a.dritschel@newcastle.ac.uk

Hugo J. Woerdeman
Affiliation: Department of Mathematics, The College of William & Mary, Williamsburg, Virginia 23185-8795 – and – Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B3001 Heverlee, Belgium
Address at time of publication: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: hugo@math.drexel.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03814-6
Received by editor(s): March 1, 2004
Published electronically: June 21, 2005
Additional Notes: The first author’s research was supported by the Engineering and Physical Sciences Research Council (EPSRC) and by the European Community’s Human Potential Programme Under Contract HPRN-CT-2000-00116 (Analysis And Operators).
The second author’s research was supported in part by the National Science Foundation (NSF), as well as a Faculty Research Assignment (FRA) Grant from the College of William & Mary.
Dedicated: In memory of Marvin Rosenblum
Article copyright: © Copyright 2005 American Mathematical Society