Outer factorization of operator valued weight functions on the torus

Cheng, Ray

Cheng, Ray

Studia Mathematica, Tome 108 (1994), p. 19-34 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $Λ = (m, n) \in ℤ^{2} : m \geq 1 \cup (0, n) : n \geq 0$ , 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.

Publié le : 1994-01-01

Zbl 0822.47021

EUDML-ID : urn:eudml:doc:216095

@article{bwmeta1.element.bwnjournal-article-smv110i1p19bwm,
     author = {Ray Cheng},
     title = {Outer factorization of operator valued weight functions on the torus},
     journal = {Studia Mathematica},
     volume = {108},
     year = {1994},
     pages = {19-34},
     zbl = {0822.47021},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv110i1p19bwm}
}

Cheng, Ray. Outer factorization of operator valued weight functions on the torus. Studia Mathematica, Tome 108 (1994) pp. 19-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv110i1p19bwm/

Bibliographie

[00000] [1] D. A. Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458-495. | Zbl 0098.08402

[00001] [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. | Zbl 0082.28201

[00002] [3] H. Korezlioglu and Ph. Loubaton, Spectral factorization of wide sense stationary processes on $ℤ^{2}$ , J. Multivariate Anal. 19 (1986), 24-47.

[00003] [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. | Zbl 0772.60028

[00004] [5] D. Lowdenslager, On factoring matrix valued functions, Ann. of Math. (2) 78 (1963), 450-454. | Zbl 0117.14701

[00005] [6] S. C. Power, Spectral characterization of the Wold-Zasuhin decomposition and prediction-error operator, Math. Proc. Cambridge Philos. Soc. 110 (1991), 559-567. | Zbl 0745.60032

[00006] [7] M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139-147. | Zbl 0159.43102

[00007] [8] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, New York, 1985. | Zbl 0586.47020

[00008] [9] Yu. A. Rozanov, Stationary Random Processes, Holden-Day, San Francisco, 1967.

[00009] [10] G. Szegö, Über die Randwerte analytischer Funktionen, Math. Ann. 84 (1921), 232-244. | Zbl 48.0332.03

[00010] [11] N. Wiener and E. J. Akutowicz, A factorization of positive Hermitian matrices, J. Math. Mech. 8 (1959), 111-120. | Zbl 0082.28103

[00011] [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. | Zbl 0080.13002