Translation-invariant Operators in Reproducing Kernel Hilbert Spaces

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.