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
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by V. Mandrekar and H. Salehi

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

DOI: https://doi.org/10.1090/S0002-9939-1972-0287350-X

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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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