Integral representation of translation-invariant operators on reproducing kernel Hilbert spaces

TL;DR

The article develops a general framework for translation-invariant operators on reproducing kernel Hilbert spaces $H$ over a product domain $G\times Y$, with $G$ a locally compact abelian group. Through fiberization via the horizontal Fourier transform, the commutant $\mathcal{C}(\rho)$ is shown to be a commutative von Neumann algebra isomorphic to $L^{\infty}(\Omega)$, and each operator is realized as an integral operator with kernel $\psi$ in a carefully defined class $\mathcal{A}$. The authors establish a full W*-algebra structure on $\mathcal{A}$, with explicit bijections linking $L^{\infty}(\Omega)$, $\mathcal{A}$, and $\mathcal{C}(\rho)$, and they derive integral representations for a broad array of special operator classes, including vertical, radial, angular, Toeplitz, and wavelet-related operators in Bergman, harmonic Bergman, and Fock spaces. The results unify and extend known C*-algebras of analytic functions on domains and provide a practical framework for spectral analysis and operator-algebraic structure in translation-invariant RKHS settings. Overall, the paper offers a comprehensive, algebraically rich description of translation-invariant operators via integral kernels and fiber decompositions, with wide-ranging examples connecting RKHS theory and operator algebras.

Abstract

We suppose that $G$ is a locally compact abelian group, $Y$ is a measure space, and $H$ is a reproducing kernel Hilbert space on $G\times Y$ such that $H$ is naturally embedded into $L^2(G\times Y)$ and it is invariant under the translations associated with $G$. We consider the von Neumann algebra of all bounded linear operators acting on $H$ that commute with these translations. Assuming that this algebra is commutative, we represent its elements as integral operators and characterize the corresponding integral kernels. Furthermore, we give W*-algebra structure on the functions associated with the integral kernels. We apply this general scheme to a series of examples, including rotation- or translation-invariant operators in Bergman or Fock spaces.

Figures (3)

• Figure 1: $V_b$ and $M_b$.
• Figure 2: Image of $K_{x,y}$ with respect to $\mathbf{F}$ and $R$.
• Figure 3: Spaces $L^\infty(\Omega)$, $\mathcal{A}$, and $\mathcal{C}(\rho)$, and bijections between them.

Theorems & Definitions (87)

• Remark 2.1: dot notation for functions with fixed arguments
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Example 2.5
• Proposition 3.1
• proof