Translation-invariant Operators in Reproducing Kernel Hilbert Spaces

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× Y, such that H is naturally embedded into L²(G× Y) and is invariant under the translations associated with the elements of G. Under some additional technical assumptions, we study the W*-algebra V of translation-invariant bounded linear operators acting on H. First, we decompose V into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces H^ _ξ, ξ∈ G^ , generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of V. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to V, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.