  Quantum Bernoulli noises (QBN, for short) are the family of annihilation and
creation operators acting on Bernoulli functionals, which satisfy a canonical
anti-commutation relation (CAR) 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 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 are shown of these quantum Markov semigroups, and examples are also
given.
