Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semi- group on the entire algebra B(h) of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractiv- ity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.