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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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.

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Keywords: Hilbert space, operator-valued functions and measures, measurability, generalized inverse factorization problem
Article copyright: © Copyright 1972 American Mathematical Society