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

Aurelian Gheondea

TL;DR

The paper develops a comprehensive theory of localised operator-valued kernels on bundles of Hilbert (and Krein) spaces that are invariant under actions of $*$-semigroupoids. It establishes generalized and bounded $*$-representations of the semigroupoids on both reproducing kernel Hilbert spaces and their minimal linearisations, with precise boundedness criteria and automatic boundedness results in the inverse semigroupoid case. The Hermitian/Krein-space extension analyzes when reproducing kernel Krein spaces exist, how generalized Krein-space linearisations arise, and how bounded representations can be obtained under strengthened conditions. The work provides a self-contained framework linking Kolmogorov-type decompositions, RKHS/KRKS theory, and semigroupoid representation theory, with potential applications to localized kernel methods and operator-valued invariant structures.

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.

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

TL;DR

The paper develops a comprehensive theory of localised operator-valued kernels on bundles of Hilbert (and Krein) spaces that are invariant under actions of -semigroupoids. It establishes generalized and bounded -representations of the semigroupoids on both reproducing kernel Hilbert spaces and their minimal linearisations, with precise boundedness criteria and automatic boundedness results in the inverse semigroupoid case. The Hermitian/Krein-space extension analyzes when reproducing kernel Krein spaces exist, how generalized Krein-space linearisations arise, and how bounded representations can be obtained under strengthened conditions. The work provides a self-contained framework linking Kolmogorov-type decompositions, RKHS/KRKS theory, and semigroupoid representation theory, with potential applications to localized kernel methods and operator-valued invariant structures.

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.
Paper Structure (17 sections, 22 theorems, 154 equations)

This paper contains 17 sections, 22 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Positive Semidefinite Operator Valued Kernels
  3. Notation and Basic Definitions.
  4. Hilbert Space Linearisations.
  5. Reproducing Kernel Hilbert Spaces.
  6. Positive Semidefinite Kernels Invariant under Actions of $*$-Semigroupoids
  7. $*$-Semigroupoids
  8. Representations of $*$-Semigroupoids
  9. Generalised $\Gamma$-Invariant Hilbert Space Linearisations
  10. $\Gamma$-Invariant Hilbert Space Linearisations.
  11. Hermitian Kernels
  12. Krein Spaces and Their Linear Operators
  13. Krein Spaces Induced by Selfadjoint Operators
  14. Krein Space Linearisations of Hermitian Kernels
  15. Reproducing Kernel Krein Spaces
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given an arbitrary bundle of Hilbert spaces $X\ast \boldsymbol{H}$ and an $\boldsymbol{H}$-operator valued kernel $K$, the following assertions are equivalent. In addition, if $K$ is positive semidefinite then a minimal Hilbert space linearisation $({\mathcal{K}};V)$ exists and it is unique, modulo unitary equivalence, that is, for any other minimal Hilbert space linearisation $({\mathcal{K}}';V'

Theorems & Definitions (52)

  • Theorem 2.1
  • proof : Proof of the implication (a)$\Rightarrow$(b) in Theorem \ref{['t:kolmogorov']}
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Example 3.1
  • ...and 42 more