Translation-invariant operators in reproducing kernel Hilbert spaces

TL;DR

This work develops a unified framework to study translation-invariant operators on reproducing kernel Hilbert spaces over product domains $G\times Y$, where $G$ is a locally compact abelian group. By applying the Fourier transform in the $G$-variable, the authors obtain a direct integral decomposition of the centralizer $\mathcal{V}$ of the translation representation, $\mathcal{V}=\rho_H'$, into fiber algebras $\mathcal{B}(\widehat{H}_\xi)$, with a concrete commutativity criterion in terms of fiber dimensions and kernel data. In the commutative case (1-dimensional fibers), they construct a unitary diagonalization $R$ that reduces $\mathcal{V}$ to multiplication by functions on the frequency set, and provide explicit formulas for the spectral function and Berezin transform; in the noncommutative finite-fiber case, they obtain a matrix- valued diagonalization. The paper also presents a Stone–Weierstrass-type result for W*-algebras and demonstrates the theory through nine examples, including vertical and radial Toeplitz operators on Bergman-type spaces and wavelet spaces. Together these results extend Vasilevski’s and related works, offering a practical toolkit for analyzing group-invariant operators in RKHS without requiring analytic or differential structure on the domain.

Abstract

Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $\mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $\mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $\widehat{H}_ξ$, $ξ\in\widehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $\mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $\mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.

Figures (1)

• Figure 1: Operators participating in Theorem \ref{['thm:R_diagonalizes_invariant_operators']}.

Theorems & Definitions (94)

• Proposition 2.1
• Theorem 2.2
• proof
• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• Corollary 4.3