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Complete Nevanlinna-Pick kernels and the characteristic function

Bhattacharyya, T and Jindal, A (2023) Complete Nevanlinna-Pick kernels and the characteristic function. In: Advances in Mathematics, 426 .

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Official URL: https://doi.org/10.1016/j.aim.2023.109089


This note finds a new characterization of complete Nevanlinna-Pick kernels on the Euclidean unit ball. The classical theory of Sz.-Nagy and Foias about the characteristic function is extended in this note to a commuting tuple T of bounded operators satisfying the natural positivity condition of 1/k-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from Hk⊗E to Hk⊗F, factoring a certain positive operator, for suitable Hilbert spaces E and F depending on T. There is a converse, which roughly says that if a kernel k admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction (1/k-contraction where k is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain.

Item Type: Journal Article
Publication: Advances in Mathematics
Publisher: Academic Press Inc.
Additional Information: The copyright for this article belongs to the Author.
Keywords: Characteristic function; Complete Nevanlinna-Pick (CNP) kernel; Generalized Bergman kernel
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 23 Jun 2023 06:36
Last Modified: 23 Jun 2023 06:39
URI: https://eprints.iisc.ac.in/id/eprint/82023

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