Localised Operator Valued Kernels Invariant under Actions of $*$-Semigroupoids

TL;DR

This paper studies operator-valued kernels invariant under $*$-semigroupoid actions, establishing their representation theory, boundedness conditions, and implications for reproducing kernel Krein spaces.

Contribution

It introduces a framework for positive and Hermitian operator-valued kernels invariant under $*$-semigroupoids, characterizes their $*$-representations, and explores boundedness and Krein space constructions.

Findings

Existence of generalized $*$-representations for invariant kernels.

Characterization of bounded $*$-representations, especially for inverse semigroupoids.

Conditions under which kernels produce reproducing kernel Krein spaces.

Abstract

We consider positive semidefinite kernels which have values given by bounded linear operators on certain bundles of Hilbert spaces and which are invariant under actions of $*$-semigroupoids. For these kernels, we prove that there exist generalised $*$-representations of the $*$-semigroupoids on the underlying reproducing kernel Hilbert spaces or, equivalently, on the underlying minimal linearisations, we characterise when the $*$-representations are performed by means of bounded operators and show that this always happens for inverse semigroupoids. Then, we consider Hermitian kernels which have values given by bounded linear operators on certain bundles of Hilbert spaces and which are invariant under actions of $*$-semigroupoids. Only those Hermitian kernels having certain boundedness properties can produce reproducing kernel Krein spaces but uniqueness is more complicated. However, for these kernels, generalised $*$-representations can be obtained. If $*$-representations with bounded linear operators are requested, then stronger boundedness conditions on the kernels are needed.