On a factorization theorem of D. Lowdenslager

Mandrekar, V.; Salehi, H.

doi:10.1090/S0002-9939-1972-0287350-X

On a factorization theorem of D. Lowdenslager
HTML articles powered by AMS MathViewer

by V. Mandrekar and H. Salehi PDF

Proc. Amer. Math. Soc. 31 (1972), 185-188 Request permission

Abstract:

For a positive-definite infinite-dimensional matrixvalued function $M$ defined on the unit circle a factorization theorem for $M$ in the form $M = A{A^\ast }$, where $A$ is a function with Fourier series ${\sum _{n > 0}}{A_n}{e^{in\theta }}$, is proved. The theorem, as was originally stated by D. Lowdenslager, contained an error. Based on our study concerning the completeness of the space of square-integrable operator-valued functions (not necessarily bounded) with respect to a nonnegative bounded operator-valued measure a correct proof of the factorization problem is provided. This work subsumes several known results concerning the factorization problem.

References

R. G. Douglas, On factoring positive operator functions, J. Math. Mech. 16 (1966), 119–126. MR 0209887, DOI 10.1512/iumj.1967.16.16007
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR 203464, DOI 10.1090/S0002-9939-1966-0203464-1
Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
Magnus R. Hestenes, Relative self-adjoint operators in Hilbert space, Pacific J. Math. 11 (1961), 1315–1357. MR 136996
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
David B. Lowdenslager, On factoring matrix valued functions, Ann. of Math. (2) 78 (1963), 450–454. MR 155154, DOI 10.2307/1970535
V. Mandrekar and H. Salehi, The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana Univ. Math. J. 20 (1970/71), 545–563. MR 267069, DOI 10.1512/iumj.1970.20.20045

Vector-valued weakly stationary stochastic processes and factorability of matrix-valued functions

Herman Wold, A study in the analysis of stationary time series, Almqvist & Wiksell, Stockholm, 1954. 2d ed; With an appendix by Peter Whittle. MR 0061344