Abstract
Given a quantum Markov semigroup (T t ) t≥o on B (h), with a faithful normal invariant state ρ, we associate to it the semigroup (T t ) t≥o on Hilbert-Schmidt operators on h (the L 2 (ρ) space) defined by (T t (ρ 1/4xρ 1/4) = ρ 1/4 T t (x)ρ 1/4. This allows us to use spectral theory to study the infinitesimal generator of (T t ) t≥o and deduce information on the speed of convergence to equilibrium of the given semigroup. We apply this idea to show that some quantum Markov semigroups related to birth-and-death processes converge to equilibrium exponentially rapidly in L 2 (p).
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Carbone, R. (2000). Exponential L 2-Convergence of Some Quantum Markov Semigroups Related to Birth-and-Death Processes. In: Rebolledo, R. (eds) Stochastic Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1372-7_1
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DOI: https://doi.org/10.1007/978-1-4612-1372-7_1
Publisher Name: Birkhäuser, Boston, MA
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