Spectral Analysis and Feller Property for Quantum Ornstein–Uhlenbeck Semigroups

Abstract:

A class of dynamical semigroups arising in quantum optics models of masers and lasers is investigated. The semigroups are constructed, by means of noncommutative Dirichlet forms, on the full algebra of bounded operators on a separable Hilbert space. The explicit action of their generators on a core in the domain is used to demonstrate the Feller property of the semigroups, with respect to the C ^*-subalgebra of compact operators. The Dirichlet forms are analysed and the C ²-spectrum together with eigenspaces are found. When reduced to certain maximal abelian subalgebras, the semigroups give rise to the Markov semigroups of classical Ornstein–Uhlenbeck processes on the one hand, and of classical birth-and-death processes on the other.