Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Outer factorizations in one and several variables
HTML articles powered by AMS MathViewer

by Michael A. Dritschel and Hugo J. Woerdeman PDF
Trans. Amer. Math. Soc. 357 (2005), 4661-4679 Request permission


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.
  • Gr. Arsene, Zoia Ceauşescu, and T. Constantinescu, Schur analysis of some completion problems, Linear Algebra Appl. 109 (1988), 1–35. MR 961563, DOI 10.1016/0024-3795(88)90195-4
  • 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
  • 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.
  • Raymond Cheng, Weakly and strongly outer functions on the bidisc, Michigan Math. J. 39 (1992), no. 1, 99–109. MR 1137892, DOI 10.1307/mmj/1029004458
  • 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
  • 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, DOI 10.1007/978-3-0348-5483-2_{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
  • 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
  • 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, DOI 10.1007/978-3-0348-7712-1
  • 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.
  • I. C. Gohberg, The factorization problem for operator functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1055–1082 (Russian). MR 0174994
  • Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
  • 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, DOI 10.1137/S0895479800371177
  • Pablo A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology, 2000.
  • 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
  • Murray Rosenblatt, A multi-dimensional prediction problem, Ark. Mat. 3 (1958), 407–424. MR 92332, DOI 10.1007/BF02589495
  • Marvin Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139–147. MR 227794, DOI 10.1016/0022-247X(68)90122-4
  • 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
  • Walter Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532–539. MR 151796
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47A68, 47B35, 15A48
  • Retrieve articles in all journals with MSC (2000): 47A68, 47B35, 15A48
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:
  • 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
  • MR Author ID: 183930
  • Email:
  • 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
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 4661-4679
  • MSC (2000): Primary 47A68, 47B35, 15A48
  • DOI:
  • MathSciNet review: 2156726