Outer factorization of operator valued weight functions on the torus

Ray Cheng

Studia Mathematica (1994)

  • Volume: 110, Issue: 1, page 19-34
  • ISSN: 0039-3223

Abstract

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An exact criterion is derived for an operator valued weight function W ( e i s , e i t ) on the torus to have a factorization W ( e i s , e i t ) = Φ ( e i s , e i t ) * Φ ( e i s , e i t ) , where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane Λ = ( m , n ) ∈ ℤ 2 : m ≥ 1 ∪ ( 0 , n ) : n ≥ 0 , and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space L 2 ( W ) of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators and other structures associated with W. The corresponding version of Szegö’s infimum is given.

How to cite

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Cheng, Ray. "Outer factorization of operator valued weight functions on the torus." Studia Mathematica 110.1 (1994): 19-34. <http://eudml.org/doc/216095>.

@article{Cheng1994,
abstract = {An exact criterion is derived for an operator valued weight function $W(e^\{is\},e^\{it\})$ on the torus to have a factorization $W(e^\{is\},e^\{it\}) = Φ(e^\{is\},e^\{it\})*Φ(e^\{is\},e^\{it\})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $Λ = \{(m,n) ∈ ℤ^2: m ≥ 1\} ∪ \{(0,n): n ≥ 0\}$, and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2(W)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators and other structures associated with W. The corresponding version of Szegö’s infimum is given.},
author = {Cheng, Ray},
journal = {Studia Mathematica},
keywords = {outer factorization; Toeplitz operator; prediction theory; Szegö's infimum; multivariate stationary process; operator valued weight function; operator valued Fourier coefficients; Helson-Lowdenslager halfplane; logarithmic integrability test; Toeplitz operators; Szegö’s infimum},
language = {eng},
number = {1},
pages = {19-34},
title = {Outer factorization of operator valued weight functions on the torus},
url = {http://eudml.org/doc/216095},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Cheng, Ray
TI - Outer factorization of operator valued weight functions on the torus
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 1
SP - 19
EP - 34
AB - An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it}) = Φ(e^{is},e^{it})*Φ(e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $Λ = {(m,n) ∈ ℤ^2: m ≥ 1} ∪ {(0,n): n ≥ 0}$, and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2(W)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators and other structures associated with W. The corresponding version of Szegö’s infimum is given.
LA - eng
KW - outer factorization; Toeplitz operator; prediction theory; Szegö's infimum; multivariate stationary process; operator valued weight function; operator valued Fourier coefficients; Helson-Lowdenslager halfplane; logarithmic integrability test; Toeplitz operators; Szegö’s infimum
UR - http://eudml.org/doc/216095
ER -

References

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  1. [1] D. A. Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458-495. Zbl0098.08402
  2. [2] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables I, II, Acta Math. 99 (1958), 165-202; ibid. 106 (1961), 175-213. Zbl0082.28201
  3. [3] H. Korezlioglu and Ph. Loubaton, Spectral factorization of wide sense stationary processes on ℤ 2 , J. Multivariate Anal. 19 (1986), 24-47. 
  4. [4] Ph. Loubaton, A regularity criterion for lexicographical prediction of multivariate wide-sense stationary processes on ℤ 2 with non-full-rank spectral densities, J. Funct. Anal. 104 (1992), 198-228. Zbl0772.60028
  5. [5] D. Lowdenslager, On factoring matrix valued functions, Ann. of Math. (2) 78 (1963), 450-454. Zbl0117.14701
  6. [6] S. C. Power, Spectral characterization of the Wold-Zasuhin decomposition and prediction-error operator, Math. Proc. Cambridge Philos. Soc. 110 (1991), 559-567. Zbl0745.60032
  7. [7] M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139-147. Zbl0159.43102
  8. [8] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, New York, 1985. Zbl0586.47020
  9. [9] Yu. A. Rozanov, Stationary Random Processes, Holden-Day, San Francisco, 1967. 
  10. [10] G. Szegö, Über die Randwerte analytischer Funktionen, Math. Ann. 84 (1921), 232-244. Zbl48.0332.03
  11. [11] N. Wiener and E. J. Akutowicz, A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120. Zbl0082.28103
  12. [12] N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes I, II, Acta Math. 98 (1957), 111-150; ibid. 99 (1958), 93-137. Zbl0080.13002

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