Localization of quantum Bernoulli noises
The family (∂{sub k},∂{sub k}{sup *}){sub k≥0} of annihilation and creation operators acting on square integrable functionals of a Bernoulli process Z= (Z{sub k}){sub k⩾0} can be interpreted as quantum Bernoulli noises. In this note we consider the operator family (ℓ{sub k},ℓ{sub k}{sup *}){sub k≥0}, where ℓ{sub k}=∂{sub k}E{sub k} with E{sub k} being the conditional expectation (operator) given σ-field σ(Z{sub j}; 0 ⩽j⩽k). We show that ℓ{sub k} (resp. ℓ{sub k}{sup *}) is essentially a kind of localization of the annihilation operator ∂{sub k} (resp. creation operator ∂{sub k}{sup *}). We examine properties of the family (ℓ{sub k},ℓ{sub k}{sup *}){sub k≥0} and prove, among other things, that ℓ{sub k} and ℓ{sub k}{sup *} satisfy a local canonical anti-communication relation and (ℓ{sub k}{sup *}){sub k≥0} forms a mutually orthogonal operator sequence although each ℓ{sub k} is not a projection operator. We find that the operator series Σ{sub k=0}{sup ∞}ℓ{sub k}{sup *}Xℓ{sub k} converges in the strong operator topology for each bounded operator X acting on square integrable functionals of Z. In particular we get an explicit sum of the operator series Σ{sub k=0}{sup ∞}ℓ{sub k}{sup *}ℓ{sub k}. A useful norm estimate on Σ{sub k=0}{sup ∞}ℓ{sub k}{sup *}Xℓ{sub k} is also obtained. Finally we show applications of our main results to quantum dynamical semigroups and quantum probability.