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On the essential selfadjointness of Dirichlet operators on group-valued path space

Author: Ernesto Acosta
Journal: Proc. Amer. Math. Soc. 122 (1994), 581-590
MSC: Primary 58G32; Secondary 58D20, 60B15
MathSciNet review: 1195711
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Abstract: Let G be a compact Lie group with Lie algebra $ \mathcal{G}$. Consider the Wiener measure P on the space

$\displaystyle {W_G} = \{ g:[0,1] \to G,g\,{\text{continuous}},g(0) = e\} $

For each h in the Cameron-Martin space H over $ \mathcal{G}$, let $ {\partial _h}$ be the associated right invariant vector field over $ {W_G}$ and let $ \partial _h^ \ast $ be its adjoint with respect to P. We prove for a particular h that the space of functions on $ {W_G}$ generated by $ {C^\infty }$-cylindrical functions on $ {W_G}$ together with one Gaussian random variable is a core for the Dirichlet operator $ \partial _h^ \ast {\partial _h}$. This is the first step in proving the essential selfadjointness of the Number operator over group-valued path spaces in the natural presumed core.

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