FIRST ORDER PERTURBATIONS OF DIRICHLET OPERATORS - EXISTENCE AND UNIQUENESS

摘要

We study perturbations of type B . V of Dirichlet operators (L(0), DL(0))) associated with Dirichlet forms of type g(0)=(u, upsilon)=1/2 integral [del(u), del(upsilon)](H) d mu on L(2)(E, mu) where E is a Finite or infinite dimensional Banach space. Here H denotes a Hilbert space densely and continuously embedded in E, Assuming qu.si-regularity of (g(0), D(g(0)) we show that there always exists a closed extension of Lu:=L(0)u+[B, del u](H) that generates a sub-Markovian C-0-semigroup of contractions on L(2)(E, mu) (resp. L(1)(E, mu)), if B is an element of L(2)(E: H, mu) and integral[B, del u](H) d mu less than or equal to 0, u greater than or equal to 0. IF D is an appropriate core for (L(0), D(L(0))) we show that there is only one closed extension of (L, D) in L(1)(E, mu) generating a strongly continuous semigroup. In particular we apply our results to operators of type Delta(H)+B .del, where Delta(H) denotes the Gross-Laplacian on an abstract Wiener space (E, H, gamma) and B=id(E)+upsilon, where upsilon takes values in the Cameron-Martin space H. (C) 1996 Academic Press, Inc. [References: 19]