Shift-Invariant Measures on Hilbert and Related Function Spaces

Abstract

We consider function spaces by analogy to Sobolev spaces and spaces of smooth functions on a finite-dimensional Euclidean space. For a real separable Hilbert space E we introduce the Hilbert space of complex-valued functions on E that are square integrable with respect to some measure λ invariant under shifts and orthogonal transformations of E. For one-parameter semigroups of selfadjoint contractions in H we obtain the strong continuity criterion and study properties of their generators. For counterparts of Sobolev spaces and spaces of smooth functions we find necessary and sufficient embedding conditions and obtain the existence conditions for traces of functions in the Sobolev spaces on hyperplanes of codimension 1 in the space E. Bibliography: 13 titles.