Vector-Valued Reproducing Kernel Hilbert $$C^*$$-Modules

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.