Journal of Applied Mathematics and Physics
Vol.03 No.07(2015), Article ID:57645,5 pages
10.4236/jamp.2015.37096
Localization of Unbounded Operators on Guichardet Spaces
Jihong Zhang, Caishi Wang, Lina Tian
School of Mathematics, Lanzhou City University, Lanzhou, Gansu, China
Email: zhjhzhangjihong@163.com
Received 3 March 2015; accepted 23 June 2015; published 30 June 2015
ABSTRACT
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 ). We examine that
and
satisfy a local CAR (canonical ani-communi- cation 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.
Keywords:
Stochastic Gradient Operator, Skorohod Integral Operator, Localization, Ex-Ponential Vector, Guichardet Spaces
1. 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, where
with
being the conditional expectation (operator). The motivation for this study comes from the following observations. It is known that
is a projection operator on Guichardet Spaces. Hence, restricted to the range of
,
coincides with the stochastic gradient operator
. We show that
(resp.
) is essentially a kind of localization of the stochastic gradient operators (resp. Skorohod integral operators
). We examine that
and
can be called a local stochastic gradient operators (resp. local Skorohod integral operators
). Then, it is necessary and important to study a pair of operator
.
This paper is organized as follows. In Section 2, we fix some necessary notations and recall main notions and facts about unbounded operators on Guichardet spaces. In Section 3, Section 4 and Section 5, we state our main results. We first examined that and
satisfy a local CAR (canonical anti-communication relation) and
forms a mutually orthogonal operator sequence although each’s 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.
2. 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
the finite power set of
, namely
where denotes the cardinality of
as a set, with
denoting the collection of
element subsets. Obviously,
. Particularly, let
be an atom of measure 1. We denote by
the usual space of square integral real-valued functions on
.
Fixing a complex separable Hilbert space, Guichardet space tensor product
, which we identify with the space of square-integrable functions
, and is denoted by
. Guichardet space enjoys a continuous tensor product structure: for each
the map
where,
.
For a Hilbert space-valued map, let
be the map
given by
when, we call
is Skorohod integrable,
is Skorohod integral operator on
and
For a map, let
and
be the maps
given by
when, we call
and
the stochastic gradient of f and the adapted gradient of
, respectively. Moreover,
where. Obviously, if
,
, where
.
Let, the adapted projection on
is the orthogonal projection onto the closed subspace
:
Remark 2.1 As Hilbert space operators,
and
are unbounded operators.
and
are closed, densely defined operators. Especially,
is adjoint operator of
and
where is the number operator,
with maximal domain and
is identical operator.
Lemma 2.1 [1] Let and
be Skorohod integrable, if the map
is integrable, then
. (1)
Lemma 2.2 [1] Let be measurable. If
for almost every
, then
, (2)
where (1) may call the canonical-commutation relations.
3. 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
with domain
, we define an conditioned expectation operator
on
by the a.e. prescription
,
with domain
where,
,
.
Clearly, is a subspce of
, and for any
, we have
for a.a.
. Thus
is an s-adapted subspace.
Remark 3.1 If is s-adapted(i.e. for all
,
for a.a.
), then the subspaces
and
coincide, and
for
in this subspace, it follows that
Whenever belongs to
. If
is densely defined, s-adapted and
is closable, then
.
Remark 3.2 is s-adapted operator and
.
Definition 3.1 For, we call
the local stochastic gradient operator and its adjoint operator
is the local Skorohod integral operator. And operator domain of
is given by
where is operator on
.
We note that for,
hence,. Especially, when
, we have
.
Theorem 3.1 By lemma2.2, we can get the following relations
(3)
which we may call the local CAR(canonical anti-commutation relations).
Proof we note that
The next theorem shows that is not a projection operator on
.
Theorem 3.2, whenever
and
.
Proof Let with
. The following algebraic relations are evident for
,
We show that, thus
We note that for
with
, which means that the
is not mutually orthogonal. However, the theorem below shows that the local operator sequence
is mutually orthogonal.
Theorem 3.3,whenever
and
.
Proof Let and
. If
, then we can show that
, from which it follows that
Now, if, then by the result of the first step we have
This completes the proof.
Theorem 3.4 is s-adapted operator if and only if
is s-adapted operator.
Proof we know that and
is s-adapted operator. we have
for a.a.
, obviously, if
is s-adapted, then
and
for a.a.
. On the other hand, if
is s-adapted,
is also s-adapted.
4. Application to Exponential Vector Formulation of QS Calculus
Recall that in the exponential vector formulation of QS calculus, all processes are defined on a domain of the algebraic tensor product form, where
is a dense subspace of
and
which is a subset of
and
denotes the expential vector of the test function
which in Guichardet spaces given by
.
For all and a.a.
, we have
and
(4)
since, the domain of the form are s-adapted. Note the a.e. identity
where
(5)
Theorem 4.1 be an operator on
with domain of the form
. Then
is s-adapted if and only if, for all
and
:
(6)
where
Proof By definition of,
be an operator on
with domain of the form
. We note that if
is s-adapted, then
. Let
and
, by (4), for a.a.
,
and so, for a.a.,
Acknowledgements
The authors are extremely grateful to the referees for their valuable comments and suggestions on improvement of the first version of the present paper. The authors are supported by National Natural Science Foundation of China (Grant No. 11261027 and No. 11461061).
Cite this paper
Jihong Zhang,Caishi Wang,Lina Tian, (2015) Localization of Unbounded Operators on Guichardet Spaces. Journal of Applied Mathematics and Physics,03,792-796. doi: 10.4236/jamp.2015.37096
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