Construction of a family of quantum Ornstein-Uhlenbeck semigroups

For a given quasi-free state on the CCR algebra over one dimensional Hilbert space, a family of Markovian semigroups which leave the quasi-free state invariant is constructed by means of noncommutative elliptic operators and Dirichlet forms on von Neumann algebras. The generators (Dirichlet operators) of the semigroups are analyzed and the spectrums together with eigenspaces are found. When restricted to a maximal Abelian subalgebra, the semigroups are reduced to a unique Markovian semigroup of classical Ornstein-Uhlenbeck process.