Vector-valued reproducing kernel Hilbert $C^*$-modules

TL;DR

This paper develops a unified framework for scalar- and vector-valued reproducing kernel Hilbert spaces within Hilbert $C^*$-modules, exploring kernels, duality, and interpolation in this setting.

Contribution

It introduces a comprehensive approach connecting positive and negative definite kernels with reproducing kernel Hilbert $C^*$-modules, including new theorems and characterizations.

Findings

Established a link between positive definite kernels and $C^*$-modules.

Proved that negative definite kernels induce reproducing kernel Hilbert $C^*$-modules.

Provided examples illustrating the theoretical results.

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.