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 L 2 ( 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.