Weighted number operators on Bernoulli functionals and quantum exclusion semigroups

Quantum Bernoulli noises (QBN for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anticommutation relation in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number operator to be bounded. Finally, as an application of the above results, we construct a class of quantum Markov semigroups associated with the weighted number operators, which belong to the category of quantum exclusion semigroups. Some basic properties of these quantum Markov semigroups are shown and examples are given.