Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup on L-p(m), p greater than or equal to 1 and m not necessarily sigma -finite. We show under mild regularity conditions that A is a Dirichlet operator in all spaces L-q(m), q greater than or equal to p. It turns out that, in the limit q --> infinity, A satisfies the positive maximum principle. If the test functions C-c(infinity) D(A), then the positive maximum principle implies that A is a pseudo-differential operator associated with a negative definite symbol, i.e., a Levy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) on LP(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular, A on L-2(m) is a symmetric integro-differential operator associated with a negative definite symbol, then A extends to a generator of a regular (symmetric) Dirichlet form on L-2(m) with explicitly given Beurling-Deny formula.