Localization of Unbounded Operators on Guichardet Spaces

As stochastic gradient and Skorohod integral operators, ( ,δ) ∇ is an adjoint pair of unbounded operators on Guichardet Spaces. In this paper, we define an adjoint pair of operator s s ( , ) ∗   , where [ ] s s s E C = ∇  with [ ] s E C being the conditional expectation operator. We show that s 


Introduction
The quantum stochastic calculus [4] [6] developed by Hudson and Parthasarathy is essentially a noncommutative extension of classical Ito stochastic calculus.In this theory, annihilation, creation, and number operator processes in boson Fock space play the role of "quantum noises", [2] which are in continuous time.On the other hand, the quantum stochastic calculus has been extended by Hitsuda is by means of the Hitsuda-Skorohod integral of anticipative process [3] [9] and the related gradient operator of Malliavin calculus.In this noncausal formulation the action of each QS integral is defined explicitly on Fock space vectors, and the essential quantum Ito formula is seen in terms of the Skorohod isometry.
In 2002, Attal [1] unify and extend both of the above approaches on Guichardet spaces.In this note, explicitly definitions of QS integrals provided and introduced no unnatural domain limitations.Moreover, maximality of operator domains is demon-strated for these QS integrals on Guichardet spaces.
In this argument, we define an adjoint pair of operator ( , )

Unbounded Operators on Guichardet Spaces
In this section, we fix some necessary notations and recall main notions and facts about unbounded operators on Guichardet spaces.For detail formulation of unbounded operators, we refer reader to [1].Let +  be the set of all nonnegative real numbers and Γ the finite power set of +  , namely where #σ denotes the cardinality of σ as a set, with ( ) n Γ denoting the collection of n element subsets. Obviously, ∅ ∈ Γ be an atom of measure 1.We denote by 2 ( ) L Γ the usual space of square integral real-valued functions on Γ .
Fixing a complex separable Hilbert space η , Guichardet space tensor product 2 ( ) L η ⊗ Γ , which we identify with the space of square-integrable functions 2 ( ; ) L η Γ , and is denoted by F .Guichardet space enjoys a conti- nuous tensor product structure: for each For a Hilbert space-valued map : ∈ , we call f ∇ and Df the stochastic gradient of f and the adapted gradient of f , respectively.Moreover, , where 2 ( , ) where ( 1) may call the canonical-commutation relations.

Local Skorohod Integral and Stochastic Gradient Operators
In the present section we state and prove our main results.We first make some preparations.
Let be an operator on F with domain V , we define an conditioned expectation operator [ ]

and for any D [ ]
We note that for the conditional expectation operator.We show that s  (resp.s *  ) is essentially a kind of localization of the stochastic gradient operators (resp.Skorohod integral operators ).We examine that s  and s As Hilbert space operators δ , ∇ and D are unbounded operators.( , <  Remark 2.1 is an s-adapted subspace.
This completes the proof.Theorem 3.4 s  is s-adapted operator if and only if s ∇ is s-adapted operator.
s E C is s-adapted operator.wehave [ ] , ≥ .On the other hand, if s is s-adapted, s ∇ is also s-adapted.