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arXiv · 2403.16401

Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$

Poornendu Kumar·Shubham Rastogi·Raghavendra Tripathi
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Abstract

Douglas and Rudin proved that any unimodular function on the unit circle $\T$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\T$.

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Poornendu Kumar, Shubham Rastogi, Raghavendra Tripathi. 2024-03-25. Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$. https://doi.org/10.1016/j.jfa.2024.110685

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