Translation-invariant operators in reproducing kernel Hilbert spaces

TL;DR

This paper investigates the structure of translation-invariant operators in reproducing kernel Hilbert spaces over locally compact abelian groups, providing decomposition, criteria for commutativity, and diagonalization methods.

Contribution

It introduces a decomposition of the algebra of translation-invariant operators into direct integrals, offers a criterion for their commutativity, and constructs a diagonalizing unitary in the commutative case.

Findings

Decomposition of operator algebra into direct integrals over Fourier transforms

Constructive criterion for algebra commutativity

Unitary diagonalization of operators in the commutative case

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 $\widehat{H}_\xi$, $\xi\in\widehat{G}$, 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.