Properties of reproducing kernel Hilbert spaces of a group action

TL;DR

The paper investigates reproducing kernel Hilbert spaces arising from a compact group $G$ acting on a compact space $X$, introducing a kernel-based equivalence relation on $X$ and showing that equivalence classes correspond to relation-stabilizer subgroups of $G$. It proves that two points are related precisely when their kernel functions are scalar multiples, and analyzes how stabilizers and their normalizers behave under the group action to illuminate when two points lie in the same equivalence class. Through these kernel–group connections, the work lays groundwork toward a full invariant-subspace decomposition result guided by the Peter–Weyl framework and identifies several open questions and directions for extending the theory. Collectively, the results link kernel geometry, group action, and subgroup structure to advance understanding of $G$-invariant spaces of $C(X)$ and their reproducing kernels.

Abstract

In this paper, we investigate properties of a reproducing kernel Hilbert space of a group action. In particular, we introduce an equivalence relation on a compact Hausdorff space $X$, and consequently establish three equivalent definitions for when two elements are related. We also see how the equivalence classes of $X$ correspond to subgroups of the group acting transitively on $X$, which we aptly refer to as relation stabilizers.

Theorems & Definitions (47)

• Conjecture
• Lemma 2.1
• Theorem 2.2: Theorem 3.5 from H2
• Theorem 2.3: Cauchy-Schwarz Inequality
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof