Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Published electronically: June 21, 2005
MathSciNet review: 2156726
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Abstract | References | Similar Articles | Additional Information

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.


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  • 1. Gr. Arsene, Zoia Ceauşescu, and T. Constantinescu, Schur analysis of some completion problems, Linear Algebra Appl. 109 (1988), 1–35. MR 961563, 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 $X$-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, 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, 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, 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 (electronic). MR 1896811, 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 0092332
  • 17. Marvin Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139–147. MR 0227794
  • 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: http://dx.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