Abstract
We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra \({{\mathcal{B}}}(h)\) of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity 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.
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Carbone, R., Sasso, E. Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup. Probab. Theory Relat. Fields 140, 505–522 (2008). https://doi.org/10.1007/s00440-007-0073-2
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DOI: https://doi.org/10.1007/s00440-007-0073-2