Gaussian semigroups of operators in the space of Borel functions on a separable Hilbert space

The concept of a Gaussian family of Borel measures on a separable Hilbert space is introduced in the paper. Necessary and sufficient conditions are found under which a Gaussian family of measures generates a semigroup of operators on the space of complex bounded Borel functions. These conditions are expressed in the form of a system of functional equations and initial conditions for operator-valued functions on the real semi-axis. A system of differential equations is derived from the system of functional equations and it is proved that the Cauchy problem has a unique solution for it. Several examples of Gaussian semigroups of operators are given.