Non-commutative chaotic expansion of Hilbert-Schmidt operators on Fock space

Abstract

It is known, from a simple algebraic computation, that every Hilbert-Schmidt operator on the Fock space admits a Maassen-Meyer kernel. Maassen-Meyer kernels are a non-commutative extension of the usual notion of chaotic expansion of random variables. Using an extension of the non-commutative stochastic integrals which allows to define these integrals on the whole Fock space, we prove that a Hilbert-Schmidt operator on Fock space is the sum of a series of iterated non-commutative stochastic integrals with respect to the basic theree quantum noises. In this way we recover its Maassen-Meyer kernel which can be completely described from the operator itself.