Localization of Unbounded Operators on Guichardet Spaces

As stochastic gradient and Skorohod integral operators, is an adjoint pair of unbounded operators on Guichardet Spaces. In this paper, we define an adjoint pair of operator , where with being the conditional expectation operator. We show that (resp.) is essentially a kind of localization of the stochastic gradient operators (resp. Skorohod integral operators δ). We examine that and satisfy a local CAR (canonical ani-communication relation) and forms a mutually orthogonal operator sequence although each is not a projection operator. We find that is s-adapted operator if and only if is s-adapted operator. Finally we show application exponential vector formulation of QS calculus.