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On a factorization theorem of D. Lowdenslager


Authors: V. Mandrekar and H. Salehi
Journal: Proc. Amer. Math. Soc. 31 (1972), 185-188
MSC: Primary 47.40; Secondary 42.00
DOI: https://doi.org/10.1090/S0002-9939-1972-0287350-X
MathSciNet review: 0287350
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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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0287350-X
Keywords: Hilbert space, operator-valued functions and measures, measurability, generalized inverse factorization problem
Article copyright: © Copyright 1972 American Mathematical Society

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