ADMISSIBLE WHITE NOISE OPERATORS AND THEIR QUANTUM WHITE NOISE DERIVATIVES

An operator on a Fock space is considered as a non-linear and non-commutative function of annihilation and creation operators at points . The derivatives with respect to a_t and , called respectively the annihilation- and creation-derivatives, are formulated within the framework of quantum white noise theory. We prove the differentiability of an admissible white noise operator and give explicit formulae for the derivatives in terms of integral kernel operators. The qwn-derivative is a non-commutative counterpart of the Gross derivative in stochastic analysis.