Malliavin calculus and Skorohod integration for quantum stochastic processes

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space h and it is shown that they satisfy similar propert ies as the derivation and divergence operator on the Wiener space over h. T he derivation operator is then used to give sufficient conditions for the e xistence of smooth Wigner densities for pairs of operators satisfying the c anonical commutation relations. For h = L-2(R+), the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integra l for adapted integrable processes and with the noncausal quantum stochasti c integrals defined by Lindsay and Belavkin for integrable processes.