# Outer factorization of operator valued weight functions on the torus

Studia Mathematica (1994)

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

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topCheng, 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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- [5] D. Lowdenslager, On factoring matrix valued functions, Ann. of Math. (2) 78 (1963), 450-454. Zbl0117.14701
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- [7] M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139-147. Zbl0159.43102
- [8] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, New York, 1985. Zbl0586.47020
- [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. Zbl48.0332.03
- [11] N. Wiener and E. J. Akutowicz, A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120. Zbl0082.28103
- [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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