Transformation Semigroups of the Space of Functions That Are Square Integrable with respect to a Translation-Invariant Measure on a Banach Space

We examine measures on a Banach space $E$ that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the $\sigma$-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space $\mathcal{H}$ of complex-valued functions of the space $E$ of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space $\mathcal{H}$ and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space $\mathcal{H}$, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity.