This short note refines a noncommutative (nc) Oka–Weil theorem by using a characterization of free compact nc sets based on the notion of dilation hulls. A consequence of it is that any free holomorphic function can be represented as a free polynomial on each free compact nc set.
In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz-Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz-Agler functions. As an application, we will show that any regular free Herglotz-Agler functions can uniformly be approximated by regular Herglotz-Agler free polynomials.
In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz class are homeomorphic. Moreover, we give a characterization of regular free Herglotz functions. As an application, we will show that any regular free Herglotz functions can uniformly be approximated by regular Herglotz free polynomials.
This short note refines nc Oka-Weil theorem by using a characterization of free compact nc sets based on the notion of dilation hulls. A consequence of it is that any free holomorphic function can be represented as a free polynomial on each free compact nc set.
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