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Vector-Valued Reproducing Kernel Hilbert \(C^*\)-Modules

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Abstract

The aim of this paper is to present a unified framework in the setting of Hilbert \(C^*\)-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and \(C^*\)-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the \(C^*\)-algebra of adjointable operators acting on a Hilbert \(C^*\)-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert \(C^*\)-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and present an interpolation theorem. Moreover, we study some basic properties of the so-called relative reproducing kernel Hilbert \(C^*\)-modules and give a characterization of dual modules. Among other things, we prove that every conditionally negative definite kernel gives us a reproducing kernel Hilbert \(C^*\)-module and a certain map. Several examples illustrate our investigation.

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Acknowledgements

This research was supported by a grant from Ferdowsi University of Mashhad (No. 2/54679).

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  1. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, 91775, Iran

    Mohammad Sal Moslehian

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Moslehian, M.S. Vector-Valued Reproducing Kernel Hilbert \(C^*\)-Modules. Complex Anal. Oper. Theory 16, 2 (2022). https://doi.org/10.1007/s11785-021-01179-3

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