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1994 | 110 | 1 | 19-34
Tytuł artykułu

Outer factorization of operator valued weight functions on the torus

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Czasopismo
Rocznik
Tom
110
Numer
1
Strony
19-34
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-03-16
poprawiono
1993-09-03
Twórcy
autor
  • Department of Mathematics, University of Louisville, Louisville, Kentucky 40292, U.S.A.
Bibliografia
  • [1] D. A. Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458-495.
  • [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.
  • [3] H. Korezlioglu and Ph. Loubaton, Spectral factorization of wide sense stationary processes on $ℤ^2$, J. Multivariate Anal. 19 (1986), 24-47.
  • [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.
  • [5] D. Lowdenslager, On factoring matrix valued functions, Ann. of Math. (2) 78 (1963), 450-454.
  • [6] S. C. Power, Spectral characterization of the Wold-Zasuhin decomposition and prediction-error operator, Math. Proc. Cambridge Philos. Soc. 110 (1991), 559-567.
  • [7] M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139-147.
  • [8] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, New York, 1985.
  • [9] Yu. A. Rozanov, Stationary Random Processes, Holden-Day, San Francisco, 1967.
  • [10] G. Szegö, Über die Randwerte analytischer Funktionen, Math. Ann. 84 (1921), 232-244.
  • [11] N. Wiener and E. J. Akutowicz, A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120.
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv110i1p19bwm
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