A dilation theoretic approach to approximation by inner functions

Alpay, D and Bhattacharyya, T and Jindal, A and Kumar, P (2023) A dilation theoretic approach to approximation by inner functions. In: Bulletin of the London Mathematical Society, 55 (6). pp. 2840-2855.

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Abstract

Using results from the 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 (Formula presented.). 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 (Formula presented.) -contractive meromorphic functions where (Formula presented.) induces an indefinite metric on (Formula presented.) 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. © 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Item Type:	Journal Article
Publication:	Bulletin of the London Mathematical Society
Publisher:	John Wiley and Sons Ltd
Additional Information:	The copyright for this article belongs to authors.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	18 Nov 2024 21:16
Last Modified:	18 Nov 2024 21:16
URI:	http://eprints.iisc.ac.in/id/eprint/85329

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