2022/03/11 by Alpay, Daniel, Bhattacharyya, Tirthankar, Jindal, Abhay +1 · 4 citations
#30E10. Secondary: 93B28 #47A56 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47A20
paper · doi:10.48550/arxiv.2203.10936
Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of H^∞. We first prove a generalization of a result of Carathéodory. This generalization has many applications. A uniform approximation result for matrix-valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix-valued functions for whom a naturally associated kernel has finitely many negative squares. This uses the Krein-Langer factorization. Approximation results for J-contractive meromorphic functions where J induces an indefinite metric on \mathbb CN are proved using the Potapov-Ginzburg Theorem. Moreover, approximation results for holomorphic functions on the unit disc with values in certain other domains of interest are also proved.