%0 Journal Article
%T Localization of Unbounded Operators on Guichardet Spaces
%A Jihong Zhang
%A Caishi Wang
%A Lina Tian
%J Journal of Applied Mathematics and Physics
%P 792-796
%@ 2327-4379
%D 2015
%I Scientific Research Publishing
%R 10.4236/jamp.2015.37096
%X <p>
	<span>As stochastic gradient and Skorohod
integral operators, <img src=\"Edit_a070bf47-7699-49bc-8a11-c320f404aa9a.png\" alt=\"\" />is an adjoint pair of
unbounded operators on Guichardet Spaces. In this paper, we define an adjoint
pair of operator <img src=\"Edit_0b933ce8-5daa-4c96-b888-d283f0f08513.png\" alt=\"\" />, where <img src=\"Edit_602e24a1-a59b-4546-9b27-7eb1347fbe5b.png\" alt=\"\" />with <img src=\"Edit_c9ea2edb-825d-46c8-8dbe-d5c8afe89b23.png\" alt=\"\" />being the conditional expectation
operator. We show that <img src=\"Edit_37b21151-6d3d-4f62-8b80-7c7360aa07c1.png\" alt=\"\" />(resp.<img src=\"Edit_fc797813-368b-45c1-8e35-a836a5381bf9.png\" alt=\"\" />) is essentially a kind of localization of the stochastic gradient
operators (resp. Skorohod integral operators </span><b><span style=\"font-family:\" cambria=\"\" math\",\"serif\";\"=\"\">¦Ä</span></b><span style=\"font-size:10.0pt;font-family:;\" \"=\"\"></span><span>). We examine that <img src=\"Edit_de46d7e3-bcc0-4b81-8ba8-572813b559cb.png\" alt=\"\" />and<img src=\"Edit_b596b426-1a55-4c7d-b38d-2b5b0821c8c5.png\" alt=\"\" /> satisfy a local CAR
(canonical ani-communication relation) and<img src=\"Edit_38cf8e80-0bbb-4474-9a5a-112edcae3488.png\" alt=\"\" /> forms a mutually
orthogonal operator sequence although each<img src=\"Edit_905f6dc9-8ba7-4164-ad70-0bd55ea682d8.png\" alt=\"\" /> is not a projection operator.
We find that <img src=\"Edit_11884e98-00bb-45eb-af08-69cba00e1a84.png\" alt=\"\" />is s-adapted operator
if and only if<img src=\"Edit_66147424-eb2f-43d2-82c9-366567b3693b.png\" alt=\"\" /> is s-adapted operator.
Finally we show application exponential vector formulation of QS calculus.</span> 
</p>
%K Stochastic Gradient Operator
%K Skorohod Integral Operator
%K Localization
%K Ex-Ponential Vector
%K Guichardet Spaces
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=57645