Douglas and Rudin proved that any unimodular function on the unit circle can be approximated, in the essential supremum norm, by the quotients of inner functions. This result has had several far reaching consequences, for instance, it formed the basis for Axler’s 1977 result on the factorization of bounded functions on the circle. We prove a similar approximation result for the operator-valued unimodular functions. In this talk, we will describe the problem and several variants of it and sketch the proof. Our proof exploits the spectral theorem to upgrade the Douglas–Rudin type approximation result for the scalar unimodular function to the operator-valued unimodular functions. This also allows us to extend Douglas–Rudin approximation to operator-valued settings on the unit sphere as well as the boundary of polydiscs. This is based on joint work with Poornendu Kumar and Shubham Rastogi.

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